Properties

Label 2-5054-1.1-c1-0-112
Degree $2$
Conductor $5054$
Sign $1$
Analytic cond. $40.3563$
Root an. cond. $6.35266$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3.04·3-s + 4-s + 3.90·5-s − 3.04·6-s + 7-s − 8-s + 6.29·9-s − 3.90·10-s − 5.43·11-s + 3.04·12-s + 2.24·13-s − 14-s + 11.8·15-s + 16-s + 0.977·17-s − 6.29·18-s + 3.90·20-s + 3.04·21-s + 5.43·22-s + 0.640·23-s − 3.04·24-s + 10.2·25-s − 2.24·26-s + 10.0·27-s + 28-s + 6.85·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.75·3-s + 0.5·4-s + 1.74·5-s − 1.24·6-s + 0.377·7-s − 0.353·8-s + 2.09·9-s − 1.23·10-s − 1.63·11-s + 0.879·12-s + 0.623·13-s − 0.267·14-s + 3.07·15-s + 0.250·16-s + 0.237·17-s − 1.48·18-s + 0.872·20-s + 0.665·21-s + 1.15·22-s + 0.133·23-s − 0.622·24-s + 2.04·25-s − 0.440·26-s + 1.93·27-s + 0.188·28-s + 1.27·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5054\)    =    \(2 \cdot 7 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(40.3563\)
Root analytic conductor: \(6.35266\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5054} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5054,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.120929251\)
\(L(\frac12)\) \(\approx\) \(4.120929251\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 - T \)
19 \( 1 \)
good3 \( 1 - 3.04T + 3T^{2} \)
5 \( 1 - 3.90T + 5T^{2} \)
11 \( 1 + 5.43T + 11T^{2} \)
13 \( 1 - 2.24T + 13T^{2} \)
17 \( 1 - 0.977T + 17T^{2} \)
23 \( 1 - 0.640T + 23T^{2} \)
29 \( 1 - 6.85T + 29T^{2} \)
31 \( 1 - 3.00T + 31T^{2} \)
37 \( 1 - 0.824T + 37T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 + 12.6T + 47T^{2} \)
53 \( 1 + 2.05T + 53T^{2} \)
59 \( 1 + 3.71T + 59T^{2} \)
61 \( 1 - 3.56T + 61T^{2} \)
67 \( 1 - 1.14T + 67T^{2} \)
71 \( 1 + 4.58T + 71T^{2} \)
73 \( 1 + 6.31T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + 9.84T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 7.94T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.262166717519441760650064601182, −7.931136681157877554242715993545, −6.98274250833703019366857104150, −6.24210042542358179653071571459, −5.35708386446883058283458761493, −4.56336687845296520354531960307, −3.13727205280318044048315872234, −2.71456318448955604776613106825, −1.98651378832727534590570166946, −1.26117692178616481589387862483, 1.26117692178616481589387862483, 1.98651378832727534590570166946, 2.71456318448955604776613106825, 3.13727205280318044048315872234, 4.56336687845296520354531960307, 5.35708386446883058283458761493, 6.24210042542358179653071571459, 6.98274250833703019366857104150, 7.931136681157877554242715993545, 8.262166717519441760650064601182

Graph of the $Z$-function along the critical line