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Properties

Label 2-9405-1.1-c1-0-124
Degree 22
Conductor 94059405
Sign 11
Analytic cond. 75.099375.0993
Root an. cond. 8.665988.66598
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 + 3·8-s + 10-s + 11-s + 2·13-s − 16-s + 6·17-s + 19-s + 20-s − 22-s + 8·23-s + 25-s − 2·26-s + 6·29-s + 4·31-s − 5·32-s − 6·34-s − 2·37-s − 38-s − 3·40-s + 10·41-s + 4·43-s − 44-s − 8·46-s − 7·49-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s + 0.316·10-s + 0.301·11-s + 0.554·13-s − 1/4·16-s + 1.45·17-s + 0.229·19-s + 0.223·20-s − 0.213·22-s + 1.66·23-s + 1/5·25-s − 0.392·26-s + 1.11·29-s + 0.718·31-s − 0.883·32-s − 1.02·34-s − 0.328·37-s − 0.162·38-s − 0.474·40-s + 1.56·41-s + 0.609·43-s − 0.150·44-s − 1.17·46-s − 49-s + ⋯

Functional equation

Λ(s)=(9405s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9405s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 94059405    =    32⋅5⋅11⋅193^{2} \cdot 5 \cdot 11 \cdot 19
Sign: 11
Analytic conductor: 75.099375.0993
Root analytic conductor: 8.665988.66598
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9405, ( :1/2), 1)(2,\ 9405,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4772306171.477230617
L(12)L(\frac12) ≈\approx 1.4772306171.477230617
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
11 1−T 1 - T
19 1−T 1 - T
good2 1+T+pT2 1 + T + p T^{2}
7 1+pT2 1 + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1−4T+pT2 1 - 4 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−7.80887232048572540682333420947, −7.22327720234686822026420079539, −6.52535200691129039267664712399, −5.53828988498538538917364873372, −4.96830657216051678599610525118, −4.15724232120626547947827499344, −3.49732228870449175323239470061, −2.64373224217327831161992119768, −1.19373409267433679159515614244, −0.823941476270948826303287224898, 0.823941476270948826303287224898, 1.19373409267433679159515614244, 2.64373224217327831161992119768, 3.49732228870449175323239470061, 4.15724232120626547947827499344, 4.96830657216051678599610525118, 5.53828988498538538917364873372, 6.52535200691129039267664712399, 7.22327720234686822026420079539, 7.80887232048572540682333420947

Graph of the ZZ-function along the critical line