Properties

Label 2-5054-1.1-c1-0-123
Degree $2$
Conductor $5054$
Sign $-1$
Analytic cond. $40.3563$
Root an. cond. $6.35266$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s + 7-s − 8-s − 3·9-s − 10-s − 2·11-s + 5·13-s − 14-s + 16-s + 3·18-s + 20-s + 2·22-s + 23-s − 4·25-s − 5·26-s + 28-s − 6·29-s − 4·31-s − 32-s + 35-s − 3·36-s + 4·37-s − 40-s + 2·41-s − 8·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 9-s − 0.316·10-s − 0.603·11-s + 1.38·13-s − 0.267·14-s + 1/4·16-s + 0.707·18-s + 0.223·20-s + 0.426·22-s + 0.208·23-s − 4/5·25-s − 0.980·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.169·35-s − 1/2·36-s + 0.657·37-s − 0.158·40-s + 0.312·41-s − 1.21·43-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: yes
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 \( 1 + p T^{2} \)
5 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−8.008014618976216737131929221182, −7.35825234205084673565926686375, −6.35057496573507231908748199838, −5.78571082214592535399532434704, −5.23951552261149761077553251142, −3.98922552317073499143400714997, −3.14341839781157544320511988142, −2.22200582649492495113955740794, −1.35805093887822549453194225798, 0, 1.35805093887822549453194225798, 2.22200582649492495113955740794, 3.14341839781157544320511988142, 3.98922552317073499143400714997, 5.23951552261149761077553251142, 5.78571082214592535399532434704, 6.35057496573507231908748199838, 7.35825234205084673565926686375, 8.008014618976216737131929221182

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