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Properties

Label 4-1110e2-1.1-c1e2-0-13
Degree 44
Conductor 12321001232100
Sign 11
Analytic cond. 78.559778.5597
Root an. cond. 2.977142.97714
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  − 4-s + 4·5-s − 9-s − 4·11-s + 16-s + 8·19-s − 4·20-s + 11·25-s + 16·29-s + 36-s + 20·41-s + 4·44-s − 4·45-s + 10·49-s − 16·55-s − 4·59-s + 12·61-s − 64-s − 24·71-s − 8·76-s + 16·79-s + 4·80-s + 81-s − 28·89-s + 32·95-s + 4·99-s − 11·100-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.78·5-s − 1/3·9-s − 1.20·11-s + 1/4·16-s + 1.83·19-s − 0.894·20-s + 11/5·25-s + 2.97·29-s + 1/6·36-s + 3.12·41-s + 0.603·44-s − 0.596·45-s + 10/7·49-s − 2.15·55-s − 0.520·59-s + 1.53·61-s − 1/8·64-s − 2.84·71-s − 0.917·76-s + 1.80·79-s + 0.447·80-s + 1/9·81-s − 2.96·89-s + 3.28·95-s + 0.402·99-s − 1.09·100-s + ⋯

Functional equation

Λ(s)=(1232100s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1232100s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12321001232100    =    22⋅32⋅52⋅3722^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}
Sign: 11
Analytic conductor: 78.559778.5597
Root analytic conductor: 2.977142.97714
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1232100, ( :1/2,1/2), 1)(4,\ 1232100,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.9510837492.951083749
L(12)L(\frac12) ≈\approx 2.9510837492.951083749
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)
bad2C2C_2 1+T2 1 + T^{2}
3C2C_2 1+T2 1 + T^{2}
5C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
37C2C_2 1+T2 1 + T^{2}
good7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
29C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
41C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
43C22C_2^2 1+58T2+p2T4 1 + 58 T^{2} + p^{2} T^{4}
47C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
53C22C_2^2 1−102T2+p2T4 1 - 102 T^{2} + p^{2} T^{4}
59C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
61C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
67C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
71C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
73C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C22C_2^2 1−150T2+p2T4 1 - 150 T^{2} + p^{2} T^{4}
89C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
97C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
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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

−9.971073844551332353811681872502, −9.749725404846743207279226353934, −9.160381611053136683879904121489, −9.060180545722797218770930246852, −8.352491084350752996434046496005, −8.211742067526369579884699621877, −7.39301760455058688424372193608, −7.32869297130414834726423199471, −6.58168453481228483765175314013, −6.12167642077108184918041839403, −5.70138181127599563863761848194, −5.47728819040868132351069304226, −4.96806792837512902741940079197, −4.59307052288847856892768091564, −3.96482606899410842392394143289, −3.03102283596293859115236733098, −2.58009415572545439057528152636, −2.56216748649858988553186066434, −1.31811950314360201096810550197, −0.869201337974042471442074666886, 0.869201337974042471442074666886, 1.31811950314360201096810550197, 2.56216748649858988553186066434, 2.58009415572545439057528152636, 3.03102283596293859115236733098, 3.96482606899410842392394143289, 4.59307052288847856892768091564, 4.96806792837512902741940079197, 5.47728819040868132351069304226, 5.70138181127599563863761848194, 6.12167642077108184918041839403, 6.58168453481228483765175314013, 7.32869297130414834726423199471, 7.39301760455058688424372193608, 8.211742067526369579884699621877, 8.352491084350752996434046496005, 9.060180545722797218770930246852, 9.160381611053136683879904121489, 9.749725404846743207279226353934, 9.971073844551332353811681872502

Graph of the ZZ-function along the critical line