Properties

Label 2-229320-1.1-c1-0-106
Degree $2$
Conductor $229320$
Sign $-1$
Analytic cond. $1831.12$
Root an. cond. $42.7916$
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  + 5-s + 11-s + 13-s − 7·17-s + 4·19-s − 3·23-s + 25-s − 2·29-s + 6·31-s − 9·37-s + 9·41-s + 2·43-s − 2·47-s + 6·53-s + 55-s + 5·59-s − 2·61-s + 65-s − 5·67-s − 8·71-s − 7·73-s − 79-s + 14·83-s − 7·85-s + 7·89-s + 4·95-s − 97-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.301·11-s + 0.277·13-s − 1.69·17-s + 0.917·19-s − 0.625·23-s + 1/5·25-s − 0.371·29-s + 1.07·31-s − 1.47·37-s + 1.40·41-s + 0.304·43-s − 0.291·47-s + 0.824·53-s + 0.134·55-s + 0.650·59-s − 0.256·61-s + 0.124·65-s − 0.610·67-s − 0.949·71-s − 0.819·73-s − 0.112·79-s + 1.53·83-s − 0.759·85-s + 0.741·89-s + 0.410·95-s − 0.101·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(229320\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1831.12\)
Root analytic conductor: \(42.7916\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{229320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 - T \)
good11 \( 1 - T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
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

−13.30203861592243, −12.74449668735043, −12.18655064467063, −11.74469644527653, −11.37835060053214, −10.78949575851151, −10.39379349170635, −9.947776417790588, −9.290187329223837, −9.040825382828225, −8.563828557725401, −8.029629850262542, −7.355097585673546, −7.051753916883821, −6.288704086314898, −6.185024713823961, −5.437671055294540, −4.982909979700053, −4.327539992386368, −3.972576054380659, −3.269404595564418, −2.637617559957270, −2.134731414930609, −1.524525070610822, −0.8365148932946873, 0, 0.8365148932946873, 1.524525070610822, 2.134731414930609, 2.637617559957270, 3.269404595564418, 3.972576054380659, 4.327539992386368, 4.982909979700053, 5.437671055294540, 6.185024713823961, 6.288704086314898, 7.051753916883821, 7.355097585673546, 8.029629850262542, 8.563828557725401, 9.040825382828225, 9.290187329223837, 9.947776417790588, 10.39379349170635, 10.78949575851151, 11.37835060053214, 11.74469644527653, 12.18655064467063, 12.74449668735043, 13.30203861592243

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