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Properties

Label 2-6498-1.1-c1-0-40
Degree 22
Conductor 64986498
Sign 11
Analytic cond. 51.886751.8867
Root an. cond. 7.203247.20324
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 − 2·5-s + 8-s − 2·10-s + 4·11-s − 2·13-s + 16-s + 6·17-s − 2·20-s + 4·22-s + 4·23-s − 25-s − 2·26-s − 2·29-s − 4·31-s + 32-s + 6·34-s − 10·37-s − 2·40-s + 10·41-s + 4·43-s + 4·44-s + 4·46-s + 4·47-s − 7·49-s − 50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s + 1.20·11-s − 0.554·13-s + 1/4·16-s + 1.45·17-s − 0.447·20-s + 0.852·22-s + 0.834·23-s − 1/5·25-s − 0.392·26-s − 0.371·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 1.64·37-s − 0.316·40-s + 1.56·41-s + 0.609·43-s + 0.603·44-s + 0.589·46-s + 0.583·47-s − 49-s − 0.141·50-s + ⋯

Functional equation

Λ(s)=(6498s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6498s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 64986498    =    2⋅32⋅1922 \cdot 3^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 51.886751.8867
Root analytic conductor: 7.203247.20324
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 6498, ( :1/2), 1)(2,\ 6498,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.9409359762.940935976
L(12)L(\frac12) ≈\approx 2.9409359762.940935976
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
19 1 1
good5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+10T+pT2 1 + 10 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 1−4T+pT2 1 - 4 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+6T+pT2 1 + 6 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

−7.84351324830482614195159853030, −7.19583741265591956235027568701, −6.73529275791267126360213503833, −5.70322294388590675424631045525, −5.20173444424979461085997861151, −4.21125984248492975559254303317, −3.72105155927604818995516413488, −3.06380200487442055779450117560, −1.89717432081133819255264343684, −0.816282240239440914761831766450, 0.816282240239440914761831766450, 1.89717432081133819255264343684, 3.06380200487442055779450117560, 3.72105155927604818995516413488, 4.21125984248492975559254303317, 5.20173444424979461085997861151, 5.70322294388590675424631045525, 6.73529275791267126360213503833, 7.19583741265591956235027568701, 7.84351324830482614195159853030

Graph of the ZZ-function along the critical line