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Properties

Label 10-2175e5-1.1-c1e5-0-2
Degree 1010
Conductor 4.867×10164.867\times 10^{16}
Sign −1-1
Analytic cond. 1.58009×1061.58009\times 10^{6}
Root an. cond. 4.167424.16742
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 55

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 5·3-s + 2·4-s + 15·6-s − 8·7-s + 2·8-s + 15·9-s + 12·11-s − 10·12-s − 2·13-s + 24·14-s − 4·16-s − 45·18-s − 2·19-s + 40·21-s − 36·22-s − 8·23-s − 10·24-s + 6·26-s − 35·27-s − 16·28-s + 5·29-s + 2·31-s + 8·32-s − 60·33-s + 30·36-s − 16·37-s + ⋯
L(s)  = 1  − 2.12·2-s − 2.88·3-s + 4-s + 6.12·6-s − 3.02·7-s + 0.707·8-s + 5·9-s + 3.61·11-s − 2.88·12-s − 0.554·13-s + 6.41·14-s − 16-s − 10.6·18-s − 0.458·19-s + 8.72·21-s − 7.67·22-s − 1.66·23-s − 2.04·24-s + 1.17·26-s − 6.73·27-s − 3.02·28-s + 0.928·29-s + 0.359·31-s + 1.41·32-s − 10.4·33-s + 5·36-s − 2.63·37-s + ⋯

Functional equation

Λ(s)=((35⋅510⋅295)s/2 ΓC(s)5 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 5^{10} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}
Λ(s)=((35⋅510⋅295)s/2 ΓC(s+1/2)5 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 5^{10} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1010
Conductor: 35⋅510⋅2953^{5} \cdot 5^{10} \cdot 29^{5}
Sign: −1-1
Analytic conductor: 1.58009×1061.58009\times 10^{6}
Root analytic conductor: 4.167424.16742
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 55
Selberg data: (10, 35⋅510⋅295, ( :1/2,1/2,1/2,1/2,1/2), −1)(10,\ 3^{5} \cdot 5^{10} \cdot 29^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3C1C_1 (1+T)5 ( 1 + T )^{5}
5 1 1
29C1C_1 (1−T)5 ( 1 - T )^{5}
good2C2≀S5C_2 \wr S_5 1+3T+7T2+13T3+23T4+33T5+23pT6+13p2T7+7p3T8+3p4T9+p5T10 1 + 3 T + 7 T^{2} + 13 T^{3} + 23 T^{4} + 33 T^{5} + 23 p T^{6} + 13 p^{2} T^{7} + 7 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10}
7C2≀S5C_2 \wr S_5 1+8T+53T2+230T3+857T4+2434T5+857pT6+230p2T7+53p3T8+8p4T9+p5T10 1 + 8 T + 53 T^{2} + 230 T^{3} + 857 T^{4} + 2434 T^{5} + 857 p T^{6} + 230 p^{2} T^{7} + 53 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10}
11C2≀S5C_2 \wr S_5 1−12T+103T2−600T3+2829T4−10300T5+2829pT6−600p2T7+103p3T8−12p4T9+p5T10 1 - 12 T + 103 T^{2} - 600 T^{3} + 2829 T^{4} - 10300 T^{5} + 2829 p T^{6} - 600 p^{2} T^{7} + 103 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10}
13C2≀S5C_2 \wr S_5 1+2T+59T2+94T3+1461T4+1772T5+1461pT6+94p2T7+59p3T8+2p4T9+p5T10 1 + 2 T + 59 T^{2} + 94 T^{3} + 1461 T^{4} + 1772 T^{5} + 1461 p T^{6} + 94 p^{2} T^{7} + 59 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10}
17C2≀S5C_2 \wr S_5 1+57T2−22T3+1665T4−450T5+1665pT6−22p2T7+57p3T8+p5T10 1 + 57 T^{2} - 22 T^{3} + 1665 T^{4} - 450 T^{5} + 1665 p T^{6} - 22 p^{2} T^{7} + 57 p^{3} T^{8} + p^{5} T^{10}
19C2≀S5C_2 \wr S_5 1+2T+27T2+132T3+338T4+172pT5+338pT6+132p2T7+27p3T8+2p4T9+p5T10 1 + 2 T + 27 T^{2} + 132 T^{3} + 338 T^{4} + 172 p T^{5} + 338 p T^{6} + 132 p^{2} T^{7} + 27 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10}
23C2≀S5C_2 \wr S_5 1+8T+109T2+676T3+4912T4+22672T5+4912pT6+676p2T7+109p3T8+8p4T9+p5T10 1 + 8 T + 109 T^{2} + 676 T^{3} + 4912 T^{4} + 22672 T^{5} + 4912 p T^{6} + 676 p^{2} T^{7} + 109 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10}
31C2≀S5C_2 \wr S_5 1−2T+53T2+16T3+2248T4−1468T5+2248pT6+16p2T7+53p3T8−2p4T9+p5T10 1 - 2 T + 53 T^{2} + 16 T^{3} + 2248 T^{4} - 1468 T^{5} + 2248 p T^{6} + 16 p^{2} T^{7} + 53 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10}
37C2≀S5C_2 \wr S_5 1+16T+223T2+2140T3+17712T4+115136T5+17712pT6+2140p2T7+223p3T8+16p4T9+p5T10 1 + 16 T + 223 T^{2} + 2140 T^{3} + 17712 T^{4} + 115136 T^{5} + 17712 p T^{6} + 2140 p^{2} T^{7} + 223 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10}
41C2≀S5C_2 \wr S_5 1+14T+169T2+1248T3+10126T4+61444T5+10126pT6+1248p2T7+169p3T8+14p4T9+p5T10 1 + 14 T + 169 T^{2} + 1248 T^{3} + 10126 T^{4} + 61444 T^{5} + 10126 p T^{6} + 1248 p^{2} T^{7} + 169 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10}
43C2≀S5C_2 \wr S_5 1+69T2−156T3+4508T4−6568T5+4508pT6−156p2T7+69p3T8+p5T10 1 + 69 T^{2} - 156 T^{3} + 4508 T^{4} - 6568 T^{5} + 4508 p T^{6} - 156 p^{2} T^{7} + 69 p^{3} T^{8} + p^{5} T^{10}
47C2≀S5C_2 \wr S_5 1+2T+127T2+342T3+8633T4+20620T5+8633pT6+342p2T7+127p3T8+2p4T9+p5T10 1 + 2 T + 127 T^{2} + 342 T^{3} + 8633 T^{4} + 20620 T^{5} + 8633 p T^{6} + 342 p^{2} T^{7} + 127 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10}
53C2≀S5C_2 \wr S_5 1+26T+441T2+5164T3+49518T4+386260T5+49518pT6+5164p2T7+441p3T8+26p4T9+p5T10 1 + 26 T + 441 T^{2} + 5164 T^{3} + 49518 T^{4} + 386260 T^{5} + 49518 p T^{6} + 5164 p^{2} T^{7} + 441 p^{3} T^{8} + 26 p^{4} T^{9} + p^{5} T^{10}
59C2≀S5C_2 \wr S_5 1−4T+85T2−404T3+3924T4−17824T5+3924pT6−404p2T7+85p3T8−4p4T9+p5T10 1 - 4 T + 85 T^{2} - 404 T^{3} + 3924 T^{4} - 17824 T^{5} + 3924 p T^{6} - 404 p^{2} T^{7} + 85 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10}
61C2≀S5C_2 \wr S_5 1+12T+253T2+2164T3+27402T4+181576T5+27402pT6+2164p2T7+253p3T8+12p4T9+p5T10 1 + 12 T + 253 T^{2} + 2164 T^{3} + 27402 T^{4} + 181576 T^{5} + 27402 p T^{6} + 2164 p^{2} T^{7} + 253 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10}
67C2≀S5C_2 \wr S_5 1+12T+293T2+2310T3+33777T4+200494T5+33777pT6+2310p2T7+293p3T8+12p4T9+p5T10 1 + 12 T + 293 T^{2} + 2310 T^{3} + 33777 T^{4} + 200494 T^{5} + 33777 p T^{6} + 2310 p^{2} T^{7} + 293 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10}
71C2≀S5C_2 \wr S_5 1−30T+567T2−7636T3+84954T4−775260T5+84954pT6−7636p2T7+567p3T8−30p4T9+p5T10 1 - 30 T + 567 T^{2} - 7636 T^{3} + 84954 T^{4} - 775260 T^{5} + 84954 p T^{6} - 7636 p^{2} T^{7} + 567 p^{3} T^{8} - 30 p^{4} T^{9} + p^{5} T^{10}
73C2≀S5C_2 \wr S_5 1−12T+227T2−2580T3+25224T4−252152T5+25224pT6−2580p2T7+227p3T8−12p4T9+p5T10 1 - 12 T + 227 T^{2} - 2580 T^{3} + 25224 T^{4} - 252152 T^{5} + 25224 p T^{6} - 2580 p^{2} T^{7} + 227 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10}
79C2≀S5C_2 \wr S_5 1+18T+327T2+44pT3+44194T4+376580T5+44194pT6+44p3T7+327p3T8+18p4T9+p5T10 1 + 18 T + 327 T^{2} + 44 p T^{3} + 44194 T^{4} + 376580 T^{5} + 44194 p T^{6} + 44 p^{3} T^{7} + 327 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10}
83C2≀S5C_2 \wr S_5 1+2T+309T2+112T3+41844T4−8972T5+41844pT6+112p2T7+309p3T8+2p4T9+p5T10 1 + 2 T + 309 T^{2} + 112 T^{3} + 41844 T^{4} - 8972 T^{5} + 41844 p T^{6} + 112 p^{2} T^{7} + 309 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10}
89C2≀S5C_2 \wr S_5 1+22T+421T2+5760T3+72333T4+717438T5+72333pT6+5760p2T7+421p3T8+22p4T9+p5T10 1 + 22 T + 421 T^{2} + 5760 T^{3} + 72333 T^{4} + 717438 T^{5} + 72333 p T^{6} + 5760 p^{2} T^{7} + 421 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10}
97C2≀S5C_2 \wr S_5 1+20T+629T2+8228T3+136670T4+1220200T5+136670pT6+8228p2T7+629p3T8+20p4T9+p5T10 1 + 20 T + 629 T^{2} + 8228 T^{3} + 136670 T^{4} + 1220200 T^{5} + 136670 p T^{6} + 8228 p^{2} T^{7} + 629 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10}
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   L(s)=∏p ∏j=110(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.01355311778597859270131890359, −5.76210118054371517262020054989, −5.67977854501064474690570106947, −5.40533470545678818905974101564, −5.36036177362494260385261017157, −4.73401512552929285281975518636, −4.71367160765961938200506968206, −4.70530082437166756509549457811, −4.67338120914734568974097030334, −4.53246181097811265902315628208, −3.96712978625600074749006692804, −3.86896209949477103890157631348, −3.64577480205658030495224175197, −3.62075680366259142733934451502, −3.56343859013510770513136402419, −3.24936797193772198072486306143, −3.00766383161831255895532087816, −2.77032007654021612627617461231, −2.30666109723907007123194318256, −2.14254364341070777381892870074, −1.81986020046948973544620214740, −1.38585558305090380304521523032, −1.28165043048159175423304649322, −1.18436371530917250460372208290, −1.11834923788114409681819649097, 0, 0, 0, 0, 0, 1.11834923788114409681819649097, 1.18436371530917250460372208290, 1.28165043048159175423304649322, 1.38585558305090380304521523032, 1.81986020046948973544620214740, 2.14254364341070777381892870074, 2.30666109723907007123194318256, 2.77032007654021612627617461231, 3.00766383161831255895532087816, 3.24936797193772198072486306143, 3.56343859013510770513136402419, 3.62075680366259142733934451502, 3.64577480205658030495224175197, 3.86896209949477103890157631348, 3.96712978625600074749006692804, 4.53246181097811265902315628208, 4.67338120914734568974097030334, 4.70530082437166756509549457811, 4.71367160765961938200506968206, 4.73401512552929285281975518636, 5.36036177362494260385261017157, 5.40533470545678818905974101564, 5.67977854501064474690570106947, 5.76210118054371517262020054989, 6.01355311778597859270131890359

Graph of the ZZ-function along the critical line