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Properties

Label 2-7938-1.1-c1-0-95
Degree 22
Conductor 79387938
Sign 11
Analytic cond. 63.385263.3852
Root an. cond. 7.961487.96148
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 1.76·5-s + 8-s + 1.76·10-s + 6.12·11-s + 0.760·13-s + 16-s + 6.84·17-s − 1.94·19-s + 1.76·20-s + 6.12·22-s − 0.421·23-s − 1.89·25-s + 0.760·26-s − 1.46·29-s + 7.70·31-s + 32-s + 6.84·34-s − 2.88·37-s − 1.94·38-s + 1.76·40-s + 6.94·41-s − 8.66·43-s + 6.12·44-s − 0.421·46-s + 1.66·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.787·5-s + 0.353·8-s + 0.556·10-s + 1.84·11-s + 0.211·13-s + 0.250·16-s + 1.65·17-s − 0.445·19-s + 0.393·20-s + 1.30·22-s − 0.0877·23-s − 0.379·25-s + 0.149·26-s − 0.271·29-s + 1.38·31-s + 0.176·32-s + 1.17·34-s − 0.474·37-s − 0.315·38-s + 0.278·40-s + 1.08·41-s − 1.32·43-s + 0.923·44-s − 0.0620·46-s + 0.242·47-s + ⋯

Functional equation

Λ(s)=(7938s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7938s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 79387938    =    2⋅34⋅722 \cdot 3^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 63.385263.3852
Root analytic conductor: 7.961487.96148
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7938, ( :1/2), 1)(2,\ 7938,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.9440390294.944039029
L(12)L(\frac12) ≈\approx 4.9440390294.944039029
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)
bad2 1−T 1 - T
3 1 1
7 1 1
good5 1−1.76T+5T2 1 - 1.76T + 5T^{2}
11 1−6.12T+11T2 1 - 6.12T + 11T^{2}
13 1−0.760T+13T2 1 - 0.760T + 13T^{2}
17 1−6.84T+17T2 1 - 6.84T + 17T^{2}
19 1+1.94T+19T2 1 + 1.94T + 19T^{2}
23 1+0.421T+23T2 1 + 0.421T + 23T^{2}
29 1+1.46T+29T2 1 + 1.46T + 29T^{2}
31 1−7.70T+31T2 1 - 7.70T + 31T^{2}
37 1+2.88T+37T2 1 + 2.88T + 37T^{2}
41 1−6.94T+41T2 1 - 6.94T + 41T^{2}
43 1+8.66T+43T2 1 + 8.66T + 43T^{2}
47 1−1.66T+47T2 1 - 1.66T + 47T^{2}
53 1−0.225T+53T2 1 - 0.225T + 53T^{2}
59 1−1.98T+59T2 1 - 1.98T + 59T^{2}
61 1+10.3T+61T2 1 + 10.3T + 61T^{2}
67 1−6.78T+67T2 1 - 6.78T + 67T^{2}
71 1−10.7T+71T2 1 - 10.7T + 71T^{2}
73 1+0.306T+73T2 1 + 0.306T + 73T^{2}
79 1+13.4T+79T2 1 + 13.4T + 79T^{2}
83 1−3.12T+83T2 1 - 3.12T + 83T^{2}
89 1+2.60T+89T2 1 + 2.60T + 89T^{2}
97 1−3.63T+97T2 1 - 3.63T + 97T^{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

−7.77551959097459725484040511980, −6.86583229272002718500581567386, −6.33973043515108301056904068761, −5.82595567193704796624620922789, −5.11911490688705541065170348480, −4.17823050235400412435662472541, −3.65264642136979553333953062348, −2.79055364976688948434794313895, −1.73684441441446196309732902466, −1.11108933339772519787956192050, 1.11108933339772519787956192050, 1.73684441441446196309732902466, 2.79055364976688948434794313895, 3.65264642136979553333953062348, 4.17823050235400412435662472541, 5.11911490688705541065170348480, 5.82595567193704796624620922789, 6.33973043515108301056904068761, 6.86583229272002718500581567386, 7.77551959097459725484040511980

Graph of the ZZ-function along the critical line