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Properties

Label 4-105e2-1.1-c1e2-0-3
Degree 44
Conductor 1102511025
Sign 11
Analytic cond. 0.7029630.702963
Root an. cond. 0.9156570.915657
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 − 2·5-s + 4·7-s − 3·9-s − 3·16-s + 12·17-s − 2·20-s + 3·25-s + 4·28-s − 8·35-s − 3·36-s − 4·37-s + 12·41-s − 16·43-s + 6·45-s − 24·47-s + 9·49-s − 24·59-s − 12·63-s − 7·64-s + 16·67-s + 12·68-s + 16·79-s + 6·80-s + 9·81-s − 24·85-s + 12·89-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.894·5-s + 1.51·7-s − 9-s − 3/4·16-s + 2.91·17-s − 0.447·20-s + 3/5·25-s + 0.755·28-s − 1.35·35-s − 1/2·36-s − 0.657·37-s + 1.87·41-s − 2.43·43-s + 0.894·45-s − 3.50·47-s + 9/7·49-s − 3.12·59-s − 1.51·63-s − 7/8·64-s + 1.95·67-s + 1.45·68-s + 1.80·79-s + 0.670·80-s + 81-s − 2.60·85-s + 1.27·89-s + ⋯

Functional equation

Λ(s)=(11025s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(11025s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1102511025    =    32⋅52⋅723^{2} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 0.7029630.702963
Root analytic conductor: 0.9156570.915657
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 11025, ( :1/2,1/2), 1)(4,\ 11025,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0826950221.082695022
L(12)L(\frac12) ≈\approx 1.0826950221.082695022
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)
bad3C2C_2 1+pT2 1 + p T^{2}
5C1C_1 (1+T)2 ( 1 + T )^{2}
7C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
good2C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
11C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
13C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
17C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
19C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
23C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
29C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
31C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
37C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
47C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
53C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
59C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
61C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
67C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
71C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
73C22C_2^2 1−98T2+p2T4 1 - 98 T^{2} + p^{2} T^{4}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
97C22C_2^2 1−146T2+p2T4 1 - 146 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

−14.44085454038547082654550860093, −13.75691866366674629273059850121, −12.82009214947011327893698412020, −12.26108891185029541558057106982, −11.73201949777095029447336782148, −11.58250802746344809847045416066, −11.02441236922778163076569590742, −10.55467498088450060964845520155, −9.769668520548840763317123178411, −9.150575487050004757995450630884, −8.253625310433564150149755056434, −7.899835355812656162524348600013, −7.78988597368559588077758915153, −6.77936700043949431351337836028, −6.07563044325703369545735779630, −5.05964223157965452541194320082, −4.96741107106739787117608112172, −3.63401856002163203666662374263, −3.02106091710520419118583338891, −1.60623861147945947594658943182, 1.60623861147945947594658943182, 3.02106091710520419118583338891, 3.63401856002163203666662374263, 4.96741107106739787117608112172, 5.05964223157965452541194320082, 6.07563044325703369545735779630, 6.77936700043949431351337836028, 7.78988597368559588077758915153, 7.899835355812656162524348600013, 8.253625310433564150149755056434, 9.150575487050004757995450630884, 9.769668520548840763317123178411, 10.55467498088450060964845520155, 11.02441236922778163076569590742, 11.58250802746344809847045416066, 11.73201949777095029447336782148, 12.26108891185029541558057106982, 12.82009214947011327893698412020, 13.75691866366674629273059850121, 14.44085454038547082654550860093

Graph of the ZZ-function along the critical line