Properties

Label 2-29645-1.1-c1-0-14
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 + 3·3-s − 4-s − 5-s + 3·6-s − 3·8-s + 6·9-s − 10-s − 3·12-s + 4·13-s − 3·15-s − 16-s + 6·18-s + 4·19-s + 20-s − 8·23-s − 9·24-s + 25-s + 4·26-s + 9·27-s − 6·29-s − 3·30-s + 2·31-s + 5·32-s − 6·36-s − 8·37-s + 4·38-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s − 1/2·4-s − 0.447·5-s + 1.22·6-s − 1.06·8-s + 2·9-s − 0.316·10-s − 0.866·12-s + 1.10·13-s − 0.774·15-s − 1/4·16-s + 1.41·18-s + 0.917·19-s + 0.223·20-s − 1.66·23-s − 1.83·24-s + 1/5·25-s + 0.784·26-s + 1.73·27-s − 1.11·29-s − 0.547·30-s + 0.359·31-s + 0.883·32-s − 36-s − 1.31·37-s + 0.648·38-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 \( 1 - p T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
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

−15.34230598622451, −14.73429737991480, −14.27260017959469, −13.77560683790079, −13.47339340592787, −13.14304469021291, −12.27532025996541, −11.98498506234875, −11.29821585991569, −10.29914706830640, −10.01703322607418, −9.208427204871347, −8.872724478363872, −8.399439649549005, −7.855954557246424, −7.369678678465814, −6.564147255146451, −5.834184041011941, −5.213630653914079, −4.266545684921945, −4.011351935412016, −3.299051842285343, −3.088748942569854, −2.024215821089865, −1.344780598710011, 0, 1.344780598710011, 2.024215821089865, 3.088748942569854, 3.299051842285343, 4.011351935412016, 4.266545684921945, 5.213630653914079, 5.834184041011941, 6.564147255146451, 7.369678678465814, 7.855954557246424, 8.399439649549005, 8.872724478363872, 9.208427204871347, 10.01703322607418, 10.29914706830640, 11.29821585991569, 11.98498506234875, 12.27532025996541, 13.14304469021291, 13.47339340592787, 13.77560683790079, 14.27260017959469, 14.73429737991480, 15.34230598622451

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