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Properties

Label 2-29645-1.1-c1-0-3
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 − 16-s − 6·17-s + 3·18-s − 2·19-s + 20-s − 5·23-s + 25-s − 29-s − 8·31-s − 5·32-s + 6·34-s + 3·36-s + 6·37-s + 2·38-s − 3·40-s + 7·41-s − 5·43-s + 3·45-s + 5·46-s + 47-s − 50-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 − 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.458·19-s + 0.223·20-s − 1.04·23-s + 1/5·25-s − 0.185·29-s − 1.43·31-s − 0.883·32-s + 1.02·34-s + 1/2·36-s + 0.986·37-s + 0.324·38-s − 0.474·40-s + 1.09·41-s − 0.762·43-s + 0.447·45-s + 0.737·46-s + 0.145·47-s − 0.141·50-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: Trivial
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 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−15.38282145591897, −14.94646729279357, −14.18455998208005, −14.05391743373994, −13.19516773658178, −12.85882157327407, −12.25320049267898, −11.31718846707842, −11.21862926806395, −10.63945634374785, −9.914690138755252, −9.338532100972527, −8.836997907020709, −8.496335557936038, −7.771937446934203, −7.499680482385988, −6.503639900917720, −6.098919261632789, −5.214457689544565, −4.693691666653931, −3.983173038439448, −3.501501299400567, −2.401380354677353, −1.896795599301853, −0.6646433566190209, 0, 0.6646433566190209, 1.896795599301853, 2.401380354677353, 3.501501299400567, 3.983173038439448, 4.693691666653931, 5.214457689544565, 6.098919261632789, 6.503639900917720, 7.499680482385988, 7.771937446934203, 8.496335557936038, 8.836997907020709, 9.338532100972527, 9.914690138755252, 10.63945634374785, 11.21862926806395, 11.31718846707842, 12.25320049267898, 12.85882157327407, 13.19516773658178, 14.05391743373994, 14.18455998208005, 14.94646729279357, 15.38282145591897

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