Properties

Label 2-350658-1.1-c1-0-82
Degree $2$
Conductor $350658$
Sign $-1$
Analytic cond. $2800.01$
Root an. cond. $52.9151$
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 − 3·5-s + 7-s + 8-s − 3·10-s + 3·13-s + 14-s + 16-s − 4·17-s − 3·20-s − 23-s + 4·25-s + 3·26-s + 28-s + 3·29-s − 6·31-s + 32-s − 4·34-s − 3·35-s − 9·37-s − 3·40-s + 9·41-s + 3·43-s − 46-s + 7·47-s + 49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.353·8-s − 0.948·10-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.670·20-s − 0.208·23-s + 4/5·25-s + 0.588·26-s + 0.188·28-s + 0.557·29-s − 1.07·31-s + 0.176·32-s − 0.685·34-s − 0.507·35-s − 1.47·37-s − 0.474·40-s + 1.40·41-s + 0.457·43-s − 0.147·46-s + 1.02·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350658 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350658 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350658\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(2800.01\)
Root analytic conductor: \(52.9151\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{350658} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 350658,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
23 \( 1 + T \)
good5 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−12.56174043429300, −12.35699671829841, −11.97344158911993, −11.36332367366742, −11.09561251939263, −10.68980108781083, −10.41381096838773, −9.444666288547259, −9.063421138844903, −8.565681180614149, −8.144637309181169, −7.637942132382051, −7.255361435955175, −6.807124512720118, −6.200780701565431, −5.796499385838091, −5.115302238879143, −4.721883694801106, −4.032629589838776, −3.937299059499920, −3.378970184751970, −2.705543817302532, −2.143262192734121, −1.468231790750315, −0.7528347374353911, 0, 0.7528347374353911, 1.468231790750315, 2.143262192734121, 2.705543817302532, 3.378970184751970, 3.937299059499920, 4.032629589838776, 4.721883694801106, 5.115302238879143, 5.796499385838091, 6.200780701565431, 6.807124512720118, 7.255361435955175, 7.637942132382051, 8.144637309181169, 8.565681180614149, 9.063421138844903, 9.444666288547259, 10.41381096838773, 10.68980108781083, 11.09561251939263, 11.36332367366742, 11.97344158911993, 12.35699671829841, 12.56174043429300

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