Properties

Label 2-5054-1.1-c1-0-115
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2.11·3-s + 4-s − 3.59·5-s − 2.11·6-s − 7-s − 8-s + 1.45·9-s + 3.59·10-s + 3.24·11-s + 2.11·12-s + 0.613·13-s + 14-s − 7.58·15-s + 16-s − 4.69·17-s − 1.45·18-s − 3.59·20-s − 2.11·21-s − 3.24·22-s + 4.53·23-s − 2.11·24-s + 7.89·25-s − 0.613·26-s − 3.25·27-s − 28-s + 4.45·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.21·3-s + 0.5·4-s − 1.60·5-s − 0.862·6-s − 0.377·7-s − 0.353·8-s + 0.486·9-s + 1.13·10-s + 0.977·11-s + 0.609·12-s + 0.170·13-s + 0.267·14-s − 1.95·15-s + 0.250·16-s − 1.13·17-s − 0.343·18-s − 0.802·20-s − 0.460·21-s − 0.691·22-s + 0.944·23-s − 0.431·24-s + 1.57·25-s − 0.120·26-s − 0.626·27-s − 0.188·28-s + 0.826·29-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: \(1\)
Selberg data: \((2,\ 5054,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.11T + 3T^{2} \)
5 \( 1 + 3.59T + 5T^{2} \)
11 \( 1 - 3.24T + 11T^{2} \)
13 \( 1 - 0.613T + 13T^{2} \)
17 \( 1 + 4.69T + 17T^{2} \)
23 \( 1 - 4.53T + 23T^{2} \)
29 \( 1 - 4.45T + 29T^{2} \)
31 \( 1 + 6.48T + 31T^{2} \)
37 \( 1 - 5.34T + 37T^{2} \)
41 \( 1 + 1.08T + 41T^{2} \)
43 \( 1 - 5.44T + 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + 6.12T + 53T^{2} \)
59 \( 1 + 2.89T + 59T^{2} \)
61 \( 1 + 13.2T + 61T^{2} \)
67 \( 1 + 6.41T + 67T^{2} \)
71 \( 1 - 11.5T + 71T^{2} \)
73 \( 1 + 16.3T + 73T^{2} \)
79 \( 1 + 6.43T + 79T^{2} \)
83 \( 1 + 8.52T + 83T^{2} \)
89 \( 1 - 1.49T + 89T^{2} \)
97 \( 1 - 1.67T + 97T^{2} \)
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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

−7.943804175276135854389858633263, −7.37466306218266279955618555298, −6.85924983936196814105581671346, −5.95801872946194430510069494409, −4.50393998828091163128787758424, −3.96954147807527950421810919550, −3.20542517492370352127111250131, −2.55611680922922566712144181291, −1.28010251928211353701233256848, 0, 1.28010251928211353701233256848, 2.55611680922922566712144181291, 3.20542517492370352127111250131, 3.96954147807527950421810919550, 4.50393998828091163128787758424, 5.95801872946194430510069494409, 6.85924983936196814105581671346, 7.37466306218266279955618555298, 7.943804175276135854389858633263

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