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Properties

Label 4-910e2-1.1-c1e2-0-53
Degree 44
Conductor 828100828100
Sign 11
Analytic cond. 52.800352.8003
Root an. cond. 2.695622.69562
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 3·4-s + 4·5-s − 4·6-s − 4·8-s + 2·9-s − 8·10-s + 4·11-s + 6·12-s − 6·13-s + 8·15-s + 5·16-s + 8·17-s − 4·18-s + 6·19-s + 12·20-s − 8·22-s − 2·23-s − 8·24-s + 11·25-s + 12·26-s + 6·27-s − 16·30-s + 8·31-s − 6·32-s + 8·33-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.78·5-s − 1.63·6-s − 1.41·8-s + 2/3·9-s − 2.52·10-s + 1.20·11-s + 1.73·12-s − 1.66·13-s + 2.06·15-s + 5/4·16-s + 1.94·17-s − 0.942·18-s + 1.37·19-s + 2.68·20-s − 1.70·22-s − 0.417·23-s − 1.63·24-s + 11/5·25-s + 2.35·26-s + 1.15·27-s − 2.92·30-s + 1.43·31-s − 1.06·32-s + 1.39·33-s + ⋯

Functional equation

Λ(s)=(828100s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(828100s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 828100828100    =    22⋅52⋅72⋅1322^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 52.800352.8003
Root analytic conductor: 2.695622.69562
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 828100, ( :1/2,1/2), 1)(4,\ 828100,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.8961070292.896107029
L(12)L(\frac12) ≈\approx 2.8961070292.896107029
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)
bad2C1C_1 (1+T)2 ( 1 + T )^{2}
5C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
7C2C_2 1+T2 1 + T^{2}
13C2C_2 1+6T+pT2 1 + 6 T + p T^{2}
good3C22C_2^2 1−2T+2T2−2pT3+p2T4 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−4T+8T2−4pT3+p2T4 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4}
17C22C_2^2 1−8T+32T2−8pT3+p2T4 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4}
19C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
29C22C_2^2 1−54T2+p2T4 1 - 54 T^{2} + p^{2} T^{4}
31C22C_2^2 1−8T+32T2−8pT3+p2T4 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C22C_2^2 1−4T+8T2−4pT3+p2T4 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4}
43C22C_2^2 1+8T+32T2+8pT3+p2T4 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
53C22C_2^2 1−8T+32T2−8pT3+p2T4 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
71C22C_2^2 1+14T+98T2+14pT3+p2T4 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4}
73C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
79C22C_2^2 1−94T2+p2T4 1 - 94 T^{2} + p^{2} T^{4}
83C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
89C22C_2^2 1−20T+200T2−20pT3+p2T4 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4}
97C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.909019313289954528911472144695, −9.863799273715597252472957019026, −9.438155601454903147557820918142, −9.258725652792588385750377375437, −8.785464947559925019730791366086, −8.290260678100921602776673099097, −7.896124883047979078406510087824, −7.44142111514652985071167618170, −7.08275830103697089805721086852, −6.65620813428663314754936913889, −5.92669686253139785541748268046, −5.91966122199761279484072653741, −4.91436355291107972404741626140, −4.84733197437917578494953704639, −3.57838092787783101088530829958, −3.21436558443299442175368177187, −2.53101598790025368461818334848, −2.34352637770315376767693448059, −1.27832549824080320124244507217, −1.21245014815265443091380975368, 1.21245014815265443091380975368, 1.27832549824080320124244507217, 2.34352637770315376767693448059, 2.53101598790025368461818334848, 3.21436558443299442175368177187, 3.57838092787783101088530829958, 4.84733197437917578494953704639, 4.91436355291107972404741626140, 5.91966122199761279484072653741, 5.92669686253139785541748268046, 6.65620813428663314754936913889, 7.08275830103697089805721086852, 7.44142111514652985071167618170, 7.896124883047979078406510087824, 8.290260678100921602776673099097, 8.785464947559925019730791366086, 9.258725652792588385750377375437, 9.438155601454903147557820918142, 9.863799273715597252472957019026, 9.909019313289954528911472144695

Graph of the ZZ-function along the critical line