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Properties

Label 2-76230-1.1-c1-0-22
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 + 2·17-s + 4·19-s − 20-s − 8·23-s + 25-s + 2·26-s − 28-s + 4·29-s + 2·31-s − 32-s − 2·34-s + 35-s − 4·37-s − 4·38-s + 40-s + 6·41-s + 8·46-s + 8·47-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 + 0.742·29-s + 0.359·31-s − 0.176·32-s − 0.342·34-s + 0.169·35-s − 0.657·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s + 1.17·46-s + 1.16·47-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

\(L(1)\) \(\approx\) \(1.316224604\)
\(L(\frac12)\) \(\approx\) \(1.316224604\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−14.10520526055460, −13.71065036425910, −12.87658908608413, −12.37588191833546, −12.01231791261645, −11.63865290807973, −11.00129572465569, −10.36701061949683, −10.00736207257536, −9.584423647186246, −9.022450963511346, −8.399159729095301, −7.897840613742791, −7.550879189638881, −6.907730052704127, −6.459367532675164, −5.718137853685093, −5.317008689397795, −4.494571570414696, −3.847752345461843, −3.337536920064027, −2.559858357103523, −2.083760716102660, −1.059692444393491, −0.4903360735847691, 0.4903360735847691, 1.059692444393491, 2.083760716102660, 2.559858357103523, 3.337536920064027, 3.847752345461843, 4.494571570414696, 5.317008689397795, 5.718137853685093, 6.459367532675164, 6.907730052704127, 7.550879189638881, 7.897840613742791, 8.399159729095301, 9.022450963511346, 9.584423647186246, 10.00736207257536, 10.36701061949683, 11.00129572465569, 11.63865290807973, 12.01231791261645, 12.37588191833546, 12.87658908608413, 13.71065036425910, 14.10520526055460

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