Properties

Label 2-5054-1.1-c1-0-47
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 − 2.77·3-s + 4-s + 1.35·5-s − 2.77·6-s + 7-s + 8-s + 4.67·9-s + 1.35·10-s + 0.777·11-s − 2.77·12-s − 4.89·13-s + 14-s − 3.75·15-s + 16-s + 2.66·17-s + 4.67·18-s + 1.35·20-s − 2.77·21-s + 0.777·22-s + 4.50·23-s − 2.77·24-s − 3.16·25-s − 4.89·26-s − 4.65·27-s + 28-s + 0.997·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.59·3-s + 0.5·4-s + 0.605·5-s − 1.13·6-s + 0.377·7-s + 0.353·8-s + 1.55·9-s + 0.427·10-s + 0.234·11-s − 0.799·12-s − 1.35·13-s + 0.267·14-s − 0.968·15-s + 0.250·16-s + 0.645·17-s + 1.10·18-s + 0.302·20-s − 0.604·21-s + 0.165·22-s + 0.939·23-s − 0.565·24-s − 0.633·25-s − 0.960·26-s − 0.895·27-s + 0.188·28-s + 0.185·29-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: \(0\)
Selberg data: \((2,\ 5054,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.989222172\)
\(L(\frac12)\) \(\approx\) \(1.989222172\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.77T + 3T^{2} \)
5 \( 1 - 1.35T + 5T^{2} \)
11 \( 1 - 0.777T + 11T^{2} \)
13 \( 1 + 4.89T + 13T^{2} \)
17 \( 1 - 2.66T + 17T^{2} \)
23 \( 1 - 4.50T + 23T^{2} \)
29 \( 1 - 0.997T + 29T^{2} \)
31 \( 1 + 7.02T + 31T^{2} \)
37 \( 1 - 5.59T + 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + 3.93T + 47T^{2} \)
53 \( 1 + 7.53T + 53T^{2} \)
59 \( 1 - 3.36T + 59T^{2} \)
61 \( 1 + 3.75T + 61T^{2} \)
67 \( 1 + 1.09T + 67T^{2} \)
71 \( 1 - 8.04T + 71T^{2} \)
73 \( 1 + 14.4T + 73T^{2} \)
79 \( 1 - 2.44T + 79T^{2} \)
83 \( 1 - 13.9T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + 2.32T + 97T^{2} \)
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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

−7.78788980164686745876829049677, −7.34869959727892053720292983511, −6.52842018928727104531954705084, −5.85589227741625951810006048824, −5.39659800960387035369394525815, −4.76725190138950601753070797028, −4.10431165187210353318574180769, −2.83095557614716877349692504307, −1.84629107946207814480770819844, −0.76121600667834168084656643913, 0.76121600667834168084656643913, 1.84629107946207814480770819844, 2.83095557614716877349692504307, 4.10431165187210353318574180769, 4.76725190138950601753070797028, 5.39659800960387035369394525815, 5.85589227741625951810006048824, 6.52842018928727104531954705084, 7.34869959727892053720292983511, 7.78788980164686745876829049677

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