Properties

Label 2-29645-1.1-c1-0-5
Degree $2$
Conductor $29645$
Sign $-1$
Analytic cond. $236.716$
Root an. cond. $15.3855$
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 + 3·8-s − 3·9-s + 10-s + 2·13-s − 16-s + 6·17-s + 3·18-s − 4·19-s + 20-s + 4·23-s + 25-s − 2·26-s − 6·29-s + 8·31-s − 5·32-s − 6·34-s + 3·36-s − 2·37-s + 4·38-s − 3·40-s + 2·41-s − 4·43-s + 3·45-s − 4·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s − 9-s + 0.316·10-s + 0.554·13-s − 1/4·16-s + 1.45·17-s + 0.707·18-s − 0.917·19-s + 0.223·20-s + 0.834·23-s + 1/5·25-s − 0.392·26-s − 1.11·29-s + 1.43·31-s − 0.883·32-s − 1.02·34-s + 1/2·36-s − 0.328·37-s + 0.648·38-s − 0.474·40-s + 0.312·41-s − 0.609·43-s + 0.447·45-s − 0.589·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29645\)    =    \(5 \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(236.716\)
Root analytic conductor: \(15.3855\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{29645} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 29645,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 + T \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.32708921946808, −14.97155366152186, −14.31691262145494, −13.81865599275983, −13.45913527502187, −12.63208860957357, −12.27404668466910, −11.59415362186766, −10.95495519025865, −10.62615871276783, −10.01068453070475, −9.274148324933091, −8.948988088025391, −8.288322875556679, −7.976948089097669, −7.386971743985302, −6.649413460277163, −5.846077113448288, −5.416573407695094, −4.634962124632869, −4.028624992936016, −3.328130105971778, −2.700796572964999, −1.591735828643313, −0.8599508722406620, 0, 0.8599508722406620, 1.591735828643313, 2.700796572964999, 3.328130105971778, 4.028624992936016, 4.634962124632869, 5.416573407695094, 5.846077113448288, 6.649413460277163, 7.386971743985302, 7.976948089097669, 8.288322875556679, 8.948988088025391, 9.274148324933091, 10.01068453070475, 10.62615871276783, 10.95495519025865, 11.59415362186766, 12.27404668466910, 12.63208860957357, 13.45913527502187, 13.81865599275983, 14.31691262145494, 14.97155366152186, 15.32708921946808

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