Properties

Label 2-29624-1.1-c1-0-3
Degree $2$
Conductor $29624$
Sign $1$
Analytic cond. $236.548$
Root an. cond. $15.3801$
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  − 2·5-s + 7-s − 3·9-s + 4·11-s + 2·13-s + 6·17-s − 8·19-s − 25-s + 6·29-s + 8·31-s − 2·35-s + 2·37-s + 2·41-s + 4·43-s + 6·45-s − 8·47-s + 49-s − 6·53-s − 8·55-s + 6·61-s − 3·63-s − 4·65-s + 4·67-s − 8·71-s + 10·73-s + 4·77-s − 16·79-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.338·35-s + 0.328·37-s + 0.312·41-s + 0.609·43-s + 0.894·45-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 1.07·55-s + 0.768·61-s − 0.377·63-s − 0.496·65-s + 0.488·67-s − 0.949·71-s + 1.17·73-s + 0.455·77-s − 1.80·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29624\)    =    \(2^{3} \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(236.548\)
Root analytic conductor: \(15.3801\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{29624} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 29624,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.915172746\)
\(L(\frac12)\) \(\approx\) \(1.915172746\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
23 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−15.04519744846792, −14.55053095698090, −14.25960759934653, −13.77412719300944, −12.86654681661842, −12.42712117718833, −11.81119051885470, −11.48748261303917, −11.09828597118221, −10.27998180275509, −9.870376718788180, −8.935845240508248, −8.549237907012132, −8.125633930135934, −7.643987437516891, −6.742936317462425, −6.212688749657977, −5.839972046757178, −4.809592889761468, −4.355235053784653, −3.685539755949762, −3.145260822601394, −2.306639290429212, −1.330761570360522, −0.5771568425087230, 0.5771568425087230, 1.330761570360522, 2.306639290429212, 3.145260822601394, 3.685539755949762, 4.355235053784653, 4.809592889761468, 5.839972046757178, 6.212688749657977, 6.742936317462425, 7.643987437516891, 8.125633930135934, 8.549237907012132, 8.935845240508248, 9.870376718788180, 10.27998180275509, 11.09828597118221, 11.48748261303917, 11.81119051885470, 12.42712117718833, 12.86654681661842, 13.77412719300944, 14.25960759934653, 14.55053095698090, 15.04519744846792

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