Properties

Label 2-29645-1.1-c1-0-1
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s + 5-s − 2·9-s + 2·12-s + 5·13-s − 15-s + 4·16-s + 3·17-s + 2·19-s − 2·20-s − 6·23-s + 25-s + 5·27-s − 3·29-s + 4·31-s + 4·36-s + 2·37-s − 5·39-s − 12·41-s + 10·43-s − 2·45-s − 9·47-s − 4·48-s − 3·51-s − 10·52-s + 12·53-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 0.447·5-s − 2/3·9-s + 0.577·12-s + 1.38·13-s − 0.258·15-s + 16-s + 0.727·17-s + 0.458·19-s − 0.447·20-s − 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s + 0.718·31-s + 2/3·36-s + 0.328·37-s − 0.800·39-s − 1.87·41-s + 1.52·43-s − 0.298·45-s − 1.31·47-s − 0.577·48-s − 0.420·51-s − 1.38·52-s + 1.64·53-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: \(0\)
Selberg data: \((2,\ 29645,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.441875060\)
\(L(\frac12)\) \(\approx\) \(1.441875060\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−14.97937570666730, −14.55462338156133, −13.96350401265218, −13.54383874409013, −13.25899189205933, −12.42193949207614, −11.97986229072058, −11.47351111685941, −10.83705248731917, −10.18656971671524, −9.893385299132744, −9.126652771028470, −8.633917514500725, −8.173218716258097, −7.607405173531416, −6.599556868166451, −6.088836036721166, −5.603940975015073, −5.189668219693005, −4.420148788117918, −3.653615481672290, −3.253322439278666, −2.178293124014615, −1.215367169526600, −0.5516424098342373, 0.5516424098342373, 1.215367169526600, 2.178293124014615, 3.253322439278666, 3.653615481672290, 4.420148788117918, 5.189668219693005, 5.603940975015073, 6.088836036721166, 6.599556868166451, 7.607405173531416, 8.173218716258097, 8.633917514500725, 9.126652771028470, 9.893385299132744, 10.18656971671524, 10.83705248731917, 11.47351111685941, 11.97986229072058, 12.42193949207614, 13.25899189205933, 13.54383874409013, 13.96350401265218, 14.55462338156133, 14.97937570666730

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