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Properties

Label 2-76230-1.1-c1-0-40
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 $0$

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 + 6·17-s − 2·19-s − 20-s + 6·23-s + 25-s − 2·26-s − 28-s + 8·31-s + 32-s + 6·34-s + 35-s + 8·37-s − 2·38-s − 40-s + 4·43-s + 6·46-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 − 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.458·19-s − 0.223·20-s + 1.25·23-s + 1/5·25-s − 0.392·26-s − 0.188·28-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 0.169·35-s + 1.31·37-s − 0.324·38-s − 0.158·40-s + 0.609·43-s + 0.884·46-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: \(0\)
Selberg data: \((2,\ 76230,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.885738484\)
\(L(\frac12)\) \(\approx\) \(3.885738484\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.17472039120745, −13.44923345721300, −13.06696399325006, −12.47261000410153, −12.21879266595374, −11.69133947791059, −11.09642345655987, −10.67299505658423, −10.05414045779491, −9.571710486289393, −9.084410333310488, −8.254874954822499, −7.824558257055037, −7.408948178216161, −6.727798891792666, −6.258962457297096, −5.730041050541681, −4.985444663020010, −4.642697925615093, −4.002949517923710, −3.229466094980208, −2.962043144958051, −2.242553675844067, −1.251447346965602, −0.6182762447542685, 0.6182762447542685, 1.251447346965602, 2.242553675844067, 2.962043144958051, 3.229466094980208, 4.002949517923710, 4.642697925615093, 4.985444663020010, 5.730041050541681, 6.258962457297096, 6.727798891792666, 7.408948178216161, 7.824558257055037, 8.254874954822499, 9.084410333310488, 9.571710486289393, 10.05414045779491, 10.67299505658423, 11.09642345655987, 11.69133947791059, 12.21879266595374, 12.47261000410153, 13.06696399325006, 13.44923345721300, 14.17472039120745

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