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Properties

Label 2-76230-1.1-c1-0-134
Degree $2$
Conductor $76230$
Sign $-1$
Analytic cond. $608.699$
Root an. cond. $24.6718$
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 − 5-s + 7-s + 8-s − 10-s + 5·13-s + 14-s + 16-s + 5·19-s − 20-s − 9·23-s + 25-s + 5·26-s + 28-s + 6·29-s + 8·31-s + 32-s − 35-s + 8·37-s + 5·38-s − 40-s − 6·41-s − 10·43-s − 9·46-s + 49-s + 50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s + 1.38·13-s + 0.267·14-s + 1/4·16-s + 1.14·19-s − 0.223·20-s − 1.87·23-s + 1/5·25-s + 0.980·26-s + 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.169·35-s + 1.31·37-s + 0.811·38-s − 0.158·40-s − 0.937·41-s − 1.52·43-s − 1.32·46-s + 1/7·49-s + 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76230\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(608.699\)
Root analytic conductor: \(24.6718\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 76230,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 \)
good13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.27600428964790, −13.69297138172862, −13.44082916276608, −12.96396285348493, −12.01268500435097, −11.86953370682723, −11.59870915667165, −10.90298670531624, −10.29041988068026, −9.985824370350124, −9.263616896905287, −8.432385546766610, −8.150651117855662, −7.816209581161914, −6.901351121534276, −6.569691239335701, −5.842635215717828, −5.582109910526043, −4.614157563060368, −4.373528685822977, −3.728122446022046, −3.076388712363089, −2.627583648669522, −1.498265410078818, −1.231589853036329, 0, 1.231589853036329, 1.498265410078818, 2.627583648669522, 3.076388712363089, 3.728122446022046, 4.373528685822977, 4.614157563060368, 5.582109910526043, 5.842635215717828, 6.569691239335701, 6.901351121534276, 7.816209581161914, 8.150651117855662, 8.432385546766610, 9.263616896905287, 9.985824370350124, 10.29041988068026, 10.90298670531624, 11.59870915667165, 11.86953370682723, 12.01268500435097, 12.96396285348493, 13.44082916276608, 13.69297138172862, 14.27600428964790

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