Properties

Label 2-5054-1.1-c1-0-104
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 − 0.352·3-s + 4-s + 4.32·5-s − 0.352·6-s + 7-s + 8-s − 2.87·9-s + 4.32·10-s + 2.35·11-s − 0.352·12-s + 0.801·13-s + 14-s − 1.52·15-s + 16-s + 6.22·17-s − 2.87·18-s + 4.32·20-s − 0.352·21-s + 2.35·22-s − 1.77·23-s − 0.352·24-s + 13.7·25-s + 0.801·26-s + 2.06·27-s + 28-s − 5.83·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.203·3-s + 0.5·4-s + 1.93·5-s − 0.143·6-s + 0.377·7-s + 0.353·8-s − 0.958·9-s + 1.36·10-s + 0.709·11-s − 0.101·12-s + 0.222·13-s + 0.267·14-s − 0.393·15-s + 0.250·16-s + 1.51·17-s − 0.677·18-s + 0.967·20-s − 0.0768·21-s + 0.501·22-s − 0.369·23-s − 0.0719·24-s + 2.74·25-s + 0.157·26-s + 0.398·27-s + 0.188·28-s − 1.08·29-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

\(L(1)\) \(\approx\) \(4.740813060\)
\(L(\frac12)\) \(\approx\) \(4.740813060\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.352T + 3T^{2} \)
5 \( 1 - 4.32T + 5T^{2} \)
11 \( 1 - 2.35T + 11T^{2} \)
13 \( 1 - 0.801T + 13T^{2} \)
17 \( 1 - 6.22T + 17T^{2} \)
23 \( 1 + 1.77T + 23T^{2} \)
29 \( 1 + 5.83T + 29T^{2} \)
31 \( 1 - 5.12T + 31T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
41 \( 1 - 3.64T + 41T^{2} \)
43 \( 1 - 0.873T + 43T^{2} \)
47 \( 1 + 5.83T + 47T^{2} \)
53 \( 1 + 1.12T + 53T^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 + 2.72T + 61T^{2} \)
67 \( 1 + 4.58T + 67T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 + 9.23T + 73T^{2} \)
79 \( 1 - 7.04T + 79T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 + 7.29T + 89T^{2} \)
97 \( 1 + 3.05T + 97T^{2} \)
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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

−8.314517626076476499838680070337, −7.29267829314900472609553866764, −6.44957194833157577845151288784, −5.89578856850836512420273487149, −5.47371668652524841200611848065, −4.85171032509243695526079126863, −3.63426876185883318176221287604, −2.84894352905872127055051205989, −1.94329287585814635329165025848, −1.20070304511893480588138578214, 1.20070304511893480588138578214, 1.94329287585814635329165025848, 2.84894352905872127055051205989, 3.63426876185883318176221287604, 4.85171032509243695526079126863, 5.47371668652524841200611848065, 5.89578856850836512420273487149, 6.44957194833157577845151288784, 7.29267829314900472609553866764, 8.314517626076476499838680070337

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