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Properties

Label 2-7605-1.1-c1-0-95
Degree 22
Conductor 76057605
Sign 11
Analytic cond. 60.726260.7262
Root an. cond. 7.792707.79270
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 5-s + 4·7-s + 3·8-s + 10-s + 2·11-s − 4·14-s − 16-s − 2·17-s + 6·19-s + 20-s − 2·22-s + 6·23-s + 25-s − 4·28-s − 2·29-s + 10·31-s − 5·32-s + 2·34-s − 4·35-s + 2·37-s − 6·38-s − 3·40-s − 6·41-s + 10·43-s − 2·44-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.51·7-s + 1.06·8-s + 0.316·10-s + 0.603·11-s − 1.06·14-s − 1/4·16-s − 0.485·17-s + 1.37·19-s + 0.223·20-s − 0.426·22-s + 1.25·23-s + 1/5·25-s − 0.755·28-s − 0.371·29-s + 1.79·31-s − 0.883·32-s + 0.342·34-s − 0.676·35-s + 0.328·37-s − 0.973·38-s − 0.474·40-s − 0.937·41-s + 1.52·43-s − 0.301·44-s + ⋯

Functional equation

Λ(s)=(7605s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7605s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 76057605    =    32⋅5⋅1323^{2} \cdot 5 \cdot 13^{2}
Sign: 11
Analytic conductor: 60.726260.7262
Root analytic conductor: 7.792707.79270
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7605, ( :1/2), 1)(2,\ 7605,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6285215431.628521543
L(12)L(\frac12) ≈\approx 1.6285215431.628521543
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}
ppFp(T)F_p(T)
bad3 1 1
5 1+T 1 + T
13 1 1
good2 1+T+pT2 1 + T + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1−10T+pT2 1 - 10 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1+12T+pT2 1 + 12 T + p T^{2}
83 1+16T+pT2 1 + 16 T + p T^{2}
89 1−2T+pT2 1 - 2 T + p T^{2}
97 1−2T+pT2 1 - 2 T + p T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.969442651108269867364483714241, −7.42128037066559792393591215373, −6.80492318260472867560284585691, −5.61662059881354107324334466304, −4.90995605920067117221322721145, −4.46271082359165802947890777732, −3.67694005875399616389192429368, −2.53877355899573267136722580164, −1.36388059501191469691228502578, −0.847050927765040751356708691063, 0.847050927765040751356708691063, 1.36388059501191469691228502578, 2.53877355899573267136722580164, 3.67694005875399616389192429368, 4.46271082359165802947890777732, 4.90995605920067117221322721145, 5.61662059881354107324334466304, 6.80492318260472867560284585691, 7.42128037066559792393591215373, 7.969442651108269867364483714241

Graph of the ZZ-function along the critical line