Properties

Label 2-5054-1.1-c1-0-118
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 − 2.53·3-s + 4-s − 0.652·5-s − 2.53·6-s + 7-s + 8-s + 3.41·9-s − 0.652·10-s + 0.532·11-s − 2.53·12-s + 1.87·13-s + 14-s + 1.65·15-s + 16-s − 1.04·17-s + 3.41·18-s − 0.652·20-s − 2.53·21-s + 0.532·22-s − 8.82·23-s − 2.53·24-s − 4.57·25-s + 1.87·26-s − 1.04·27-s + 28-s − 4.81·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.46·3-s + 0.5·4-s − 0.291·5-s − 1.03·6-s + 0.377·7-s + 0.353·8-s + 1.13·9-s − 0.206·10-s + 0.160·11-s − 0.730·12-s + 0.521·13-s + 0.267·14-s + 0.426·15-s + 0.250·16-s − 0.252·17-s + 0.804·18-s − 0.145·20-s − 0.552·21-s + 0.113·22-s − 1.83·23-s − 0.516·24-s − 0.914·25-s + 0.368·26-s − 0.200·27-s + 0.188·28-s − 0.894·29-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 \( 1 + 2.53T + 3T^{2} \)
5 \( 1 + 0.652T + 5T^{2} \)
11 \( 1 - 0.532T + 11T^{2} \)
13 \( 1 - 1.87T + 13T^{2} \)
17 \( 1 + 1.04T + 17T^{2} \)
23 \( 1 + 8.82T + 23T^{2} \)
29 \( 1 + 4.81T + 29T^{2} \)
31 \( 1 - 5.59T + 31T^{2} \)
37 \( 1 - 6.90T + 37T^{2} \)
41 \( 1 - 2.79T + 41T^{2} \)
43 \( 1 + 7.36T + 43T^{2} \)
47 \( 1 + 0.411T + 47T^{2} \)
53 \( 1 - 1.98T + 53T^{2} \)
59 \( 1 - 2.73T + 59T^{2} \)
61 \( 1 + 6.69T + 61T^{2} \)
67 \( 1 - 3.43T + 67T^{2} \)
71 \( 1 + 0.554T + 71T^{2} \)
73 \( 1 - 0.170T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 - 5.38T + 97T^{2} \)
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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

−7.77561965244429940889764076900, −6.87312884115100455006396356450, −6.10754534088389438249603745371, −5.84721320615097025892865316409, −4.96048759842280866786746061566, −4.27394665307009153602069525368, −3.68487258486702079381383305646, −2.33506187518766204656078071439, −1.29897141774553520352100295367, 0, 1.29897141774553520352100295367, 2.33506187518766204656078071439, 3.68487258486702079381383305646, 4.27394665307009153602069525368, 4.96048759842280866786746061566, 5.84721320615097025892865316409, 6.10754534088389438249603745371, 6.87312884115100455006396356450, 7.77561965244429940889764076900

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