Properties

Label 2-29645-1.1-c1-0-13
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 $2$

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 − 4·13-s − 16-s + 3·18-s + 4·19-s + 20-s − 6·23-s + 25-s + 4·26-s − 6·31-s − 5·32-s + 3·36-s + 2·37-s − 4·38-s − 3·40-s − 2·41-s − 12·43-s + 3·45-s + 6·46-s − 8·47-s − 50-s + 4·52-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 − 1.10·13-s − 1/4·16-s + 0.707·18-s + 0.917·19-s + 0.223·20-s − 1.25·23-s + 1/5·25-s + 0.784·26-s − 1.07·31-s − 0.883·32-s + 1/2·36-s + 0.328·37-s − 0.648·38-s − 0.474·40-s − 0.312·41-s − 1.82·43-s + 0.447·45-s + 0.884·46-s − 1.16·47-s − 0.141·50-s + 0.554·52-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: \(2\)
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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.90018835779264, −14.99423293002819, −14.60990369756792, −14.19648597976459, −13.58448472033072, −13.13105464488524, −12.20465572857485, −12.11361153849788, −11.20003206771007, −10.99241097452040, −10.06206066939690, −9.646383204541294, −9.359621460486729, −8.420800476381803, −8.186612955035387, −7.677404478819901, −7.034591644785657, −6.362148047290545, −5.399405954425043, −5.134503067959207, −4.397932948797195, −3.625289779109346, −3.052200726369852, −2.121224655987091, −1.335030249677101, 0, 0, 1.335030249677101, 2.121224655987091, 3.052200726369852, 3.625289779109346, 4.397932948797195, 5.134503067959207, 5.399405954425043, 6.362148047290545, 7.034591644785657, 7.677404478819901, 8.186612955035387, 8.420800476381803, 9.359621460486729, 9.646383204541294, 10.06206066939690, 10.99241097452040, 11.20003206771007, 12.11361153849788, 12.20465572857485, 13.13105464488524, 13.58448472033072, 14.19648597976459, 14.60990369756792, 14.99423293002819, 15.90018835779264

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