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Properties

Label 2-341e2-1.1-c1-0-2
Degree $2$
Conductor $116281$
Sign $-1$
Analytic cond. $928.508$
Root an. cond. $30.4714$
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 − 2·3-s − 4-s + 5-s + 2·6-s + 2·7-s + 3·8-s + 9-s − 10-s + 2·12-s + 13-s − 2·14-s − 2·15-s − 16-s − 5·17-s − 18-s − 6·19-s − 20-s − 4·21-s − 2·23-s − 6·24-s − 4·25-s − 26-s + 4·27-s − 2·28-s + 9·29-s + 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.577·12-s + 0.277·13-s − 0.534·14-s − 0.516·15-s − 1/4·16-s − 1.21·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.872·21-s − 0.417·23-s − 1.22·24-s − 4/5·25-s − 0.196·26-s + 0.769·27-s − 0.377·28-s + 1.67·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116281\)    =    \(11^{2} \cdot 31^{2}\)
Sign: $-1$
Analytic conductor: \(928.508\)
Root analytic conductor: \(30.4714\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 116281,\ (\ :1/2),\ -1)\)

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)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−13.80592743910908, −13.29864995465893, −12.83858210260052, −12.39638962544195, −11.70864702337995, −11.28235120729512, −10.87118746154353, −10.53140598284302, −9.961937076012346, −9.554563236646717, −8.784047045406304, −8.517837293030718, −8.083885840372632, −7.475149987832276, −6.669388845019102, −6.346801749245137, −5.891702681460363, −5.113778399086025, −4.803474768249593, −4.280749104694457, −3.803934934688625, −2.587098722451806, −2.086706982810597, −1.369444458438933, −0.6822655334418399, 0, 0.6822655334418399, 1.369444458438933, 2.086706982810597, 2.587098722451806, 3.803934934688625, 4.280749104694457, 4.803474768249593, 5.113778399086025, 5.891702681460363, 6.346801749245137, 6.669388845019102, 7.475149987832276, 8.083885840372632, 8.517837293030718, 8.784047045406304, 9.554563236646717, 9.961937076012346, 10.53140598284302, 10.87118746154353, 11.28235120729512, 11.70864702337995, 12.39638962544195, 12.83858210260052, 13.29864995465893, 13.80592743910908

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