Properties

Label 2-5054-1.1-c1-0-1
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 − 2.98·3-s + 4-s − 4.31·5-s + 2.98·6-s + 7-s − 8-s + 5.89·9-s + 4.31·10-s + 0.362·11-s − 2.98·12-s − 5.24·13-s − 14-s + 12.8·15-s + 16-s − 4.62·17-s − 5.89·18-s − 4.31·20-s − 2.98·21-s − 0.362·22-s + 1.78·23-s + 2.98·24-s + 13.6·25-s + 5.24·26-s − 8.61·27-s + 28-s − 2.08·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.72·3-s + 0.5·4-s − 1.92·5-s + 1.21·6-s + 0.377·7-s − 0.353·8-s + 1.96·9-s + 1.36·10-s + 0.109·11-s − 0.860·12-s − 1.45·13-s − 0.267·14-s + 3.32·15-s + 0.250·16-s − 1.12·17-s − 1.38·18-s − 0.964·20-s − 0.650·21-s − 0.0772·22-s + 0.371·23-s + 0.608·24-s + 2.72·25-s + 1.02·26-s − 1.65·27-s + 0.188·28-s − 0.387·29-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

\(L(1)\) \(\approx\) \(0.03850555429\)
\(L(\frac12)\) \(\approx\) \(0.03850555429\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.98T + 3T^{2} \)
5 \( 1 + 4.31T + 5T^{2} \)
11 \( 1 - 0.362T + 11T^{2} \)
13 \( 1 + 5.24T + 13T^{2} \)
17 \( 1 + 4.62T + 17T^{2} \)
23 \( 1 - 1.78T + 23T^{2} \)
29 \( 1 + 2.08T + 29T^{2} \)
31 \( 1 - 2.02T + 31T^{2} \)
37 \( 1 + 7.44T + 37T^{2} \)
41 \( 1 - 2.71T + 41T^{2} \)
43 \( 1 - 3.97T + 43T^{2} \)
47 \( 1 + 3.27T + 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + 0.777T + 59T^{2} \)
61 \( 1 + 10.7T + 61T^{2} \)
67 \( 1 - 2.61T + 67T^{2} \)
71 \( 1 + 14.5T + 71T^{2} \)
73 \( 1 - 2.61T + 73T^{2} \)
79 \( 1 + 0.554T + 79T^{2} \)
83 \( 1 + 5.06T + 83T^{2} \)
89 \( 1 + 7.02T + 89T^{2} \)
97 \( 1 + 1.38T + 97T^{2} \)
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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

−8.036896799799943338357136342535, −7.37172932838293004520166670823, −7.00867446564004946543513375206, −6.28926840966983907094815028976, −5.15368811962929190593903005504, −4.66997064805901347408234201683, −4.03170868989201495864573548548, −2.80957038712972115508302692271, −1.37882925740794122536290987002, −0.13855234926885200313988789089, 0.13855234926885200313988789089, 1.37882925740794122536290987002, 2.80957038712972115508302692271, 4.03170868989201495864573548548, 4.66997064805901347408234201683, 5.15368811962929190593903005504, 6.28926840966983907094815028976, 7.00867446564004946543513375206, 7.37172932838293004520166670823, 8.036896799799943338357136342535

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