Properties

Label 2-229320-1.1-c1-0-71
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 − 13-s − 4·17-s − 6·19-s − 6·23-s + 25-s − 4·29-s − 8·31-s − 6·37-s + 6·41-s + 4·43-s + 8·47-s − 2·53-s + 2·61-s − 65-s − 4·67-s + 8·71-s + 16·79-s − 4·83-s − 4·85-s − 6·89-s − 6·95-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.277·13-s − 0.970·17-s − 1.37·19-s − 1.25·23-s + 1/5·25-s − 0.742·29-s − 1.43·31-s − 0.986·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s − 0.274·53-s + 0.256·61-s − 0.124·65-s − 0.488·67-s + 0.949·71-s + 1.80·79-s − 0.439·83-s − 0.433·85-s − 0.635·89-s − 0.615·95-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−13.06111912253740, −12.68933629183560, −12.40896181719607, −11.79128182506265, −11.19639459163950, −10.70291667027289, −10.61202388398431, −9.816767290578042, −9.448636626576803, −8.908863660169583, −8.620064281133137, −7.955027870291320, −7.478088836474440, −6.942989581615446, −6.496796082941444, −5.893317687402635, −5.611385793281580, −4.943624137507778, −4.265665887801836, −4.012665769805623, −3.348433368121441, −2.525866945341528, −2.007595279176981, −1.810855697055542, −0.6648579686446148, 0, 0.6648579686446148, 1.810855697055542, 2.007595279176981, 2.525866945341528, 3.348433368121441, 4.012665769805623, 4.265665887801836, 4.943624137507778, 5.611385793281580, 5.893317687402635, 6.496796082941444, 6.942989581615446, 7.478088836474440, 7.955027870291320, 8.620064281133137, 8.908863660169583, 9.448636626576803, 9.816767290578042, 10.61202388398431, 10.70291667027289, 11.19639459163950, 11.79128182506265, 12.40896181719607, 12.68933629183560, 13.06111912253740

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