Properties

Label 2-5054-1.1-c1-0-143
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 − 0.618·3-s + 4-s + 5-s − 0.618·6-s − 7-s + 8-s − 2.61·9-s + 10-s − 5·11-s − 0.618·12-s + 4.85·13-s − 14-s − 0.618·15-s + 16-s + 6.85·17-s − 2.61·18-s + 20-s + 0.618·21-s − 5·22-s − 3.76·23-s − 0.618·24-s − 4·25-s + 4.85·26-s + 3.47·27-s − 28-s − 8.23·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.356·3-s + 0.5·4-s + 0.447·5-s − 0.252·6-s − 0.377·7-s + 0.353·8-s − 0.872·9-s + 0.316·10-s − 1.50·11-s − 0.178·12-s + 1.34·13-s − 0.267·14-s − 0.159·15-s + 0.250·16-s + 1.66·17-s − 0.617·18-s + 0.223·20-s + 0.134·21-s − 1.06·22-s − 0.784·23-s − 0.126·24-s − 0.800·25-s + 0.951·26-s + 0.668·27-s − 0.188·28-s − 1.52·29-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: Trivial
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 \( 1 + 0.618T + 3T^{2} \)
5 \( 1 - T + 5T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 - 4.85T + 13T^{2} \)
17 \( 1 - 6.85T + 17T^{2} \)
23 \( 1 + 3.76T + 23T^{2} \)
29 \( 1 + 8.23T + 29T^{2} \)
31 \( 1 + 6.70T + 31T^{2} \)
37 \( 1 - 5.94T + 37T^{2} \)
41 \( 1 + 1.09T + 41T^{2} \)
43 \( 1 + 6.85T + 43T^{2} \)
47 \( 1 - 7.94T + 47T^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 - 11T + 59T^{2} \)
61 \( 1 - 7.47T + 61T^{2} \)
67 \( 1 - 7.85T + 67T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 + 8.85T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 - 4.85T + 89T^{2} \)
97 \( 1 + 9.47T + 97T^{2} \)
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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

−7.88715610108539321239776415825, −7.06845957627723859894693942724, −5.96780323568502429672394527833, −5.67776377120205288220024769371, −5.34975056671965840447134039218, −4.02620808779089183486358775718, −3.34746484326330225806900423569, −2.58307551410786850409947326964, −1.52007451308014621548898381245, 0, 1.52007451308014621548898381245, 2.58307551410786850409947326964, 3.34746484326330225806900423569, 4.02620808779089183486358775718, 5.34975056671965840447134039218, 5.67776377120205288220024769371, 5.96780323568502429672394527833, 7.06845957627723859894693942724, 7.88715610108539321239776415825

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