Properties

Label 2-5082-1.1-c1-0-43
Degree $2$
Conductor $5082$
Sign $1$
Analytic cond. $40.5799$
Root an. cond. $6.37024$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 2·5-s − 6-s + 7-s − 8-s + 9-s − 2·10-s + 12-s + 2·13-s − 14-s + 2·15-s + 16-s + 2·17-s − 18-s + 2·20-s + 21-s − 24-s − 25-s − 2·26-s + 27-s + 28-s + 2·29-s − 2·30-s + 4·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.447·20-s + 0.218·21-s − 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.365·30-s + 0.718·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5082 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5082 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5082\)    =    \(2 \cdot 3 \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(40.5799\)
Root analytic conductor: \(6.37024\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5082,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.439107936\)
\(L(\frac12)\) \(\approx\) \(2.439107936\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.222342273474573779535958878238, −7.76606514900733948109427526908, −6.87204263221520597540152628060, −6.15764371174594687030190015062, −5.49907117448076233936059018391, −4.51889757866883921400584450355, −3.52521205154929257820001588956, −2.61508625734212454439516390514, −1.84572904708575779361666937440, −0.973410034959001153441111592272, 0.973410034959001153441111592272, 1.84572904708575779361666937440, 2.61508625734212454439516390514, 3.52521205154929257820001588956, 4.51889757866883921400584450355, 5.49907117448076233936059018391, 6.15764371174594687030190015062, 6.87204263221520597540152628060, 7.76606514900733948109427526908, 8.222342273474573779535958878238

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