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Properties

Label 2-76230-1.1-c1-0-126
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 + 2·13-s + 14-s + 16-s + 2·17-s − 4·19-s − 20-s + 8·23-s + 25-s + 2·26-s + 28-s − 2·29-s + 32-s + 2·34-s − 35-s + 6·37-s − 4·38-s − 40-s − 6·41-s + 4·43-s + 8·46-s + 49-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 + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.223·20-s + 1.66·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.371·29-s + 0.176·32-s + 0.342·34-s − 0.169·35-s + 0.986·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s + 0.609·43-s + 1.17·46-s + 1/7·49-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.38216493234458, −13.69739987593565, −13.31595275398441, −12.85619991945890, −12.32435245996372, −11.89180700286916, −11.26469748235474, −10.94467160664893, −10.51869358012808, −9.872311680267353, −9.089574801689039, −8.722436920179549, −8.154466271675465, −7.496442836365619, −7.158087042349536, −6.514096260555471, −5.859129096738789, −5.513945923182068, −4.549698724759020, −4.505570801935610, −3.699900382854303, −3.052133823795321, −2.620149230588404, −1.608475189993222, −1.131737736794586, 0, 1.131737736794586, 1.608475189993222, 2.620149230588404, 3.052133823795321, 3.699900382854303, 4.505570801935610, 4.549698724759020, 5.513945923182068, 5.859129096738789, 6.514096260555471, 7.158087042349536, 7.496442836365619, 8.154466271675465, 8.722436920179549, 9.089574801689039, 9.872311680267353, 10.51869358012808, 10.94467160664893, 11.26469748235474, 11.89180700286916, 12.32435245996372, 12.85619991945890, 13.31595275398441, 13.69739987593565, 14.38216493234458

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