Properties

Degree $2$
Conductor $76230$
Sign $1$
Motivic weight $1$
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 + 6·13-s + 14-s + 16-s + 2·17-s + 20-s + 25-s + 6·26-s + 28-s + 6·29-s + 8·31-s + 32-s + 2·34-s + 35-s − 10·37-s + 40-s + 2·41-s − 4·43-s − 8·47-s + 49-s + 50-s + 6·52-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 + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.223·20-s + 1/5·25-s + 1.17·26-s + 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.169·35-s − 1.64·37-s + 0.158·40-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.141·50-s + 0.832·52-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: $1$
Motivic weight: \(1\)
Character: $\chi_{76230} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 76230,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.574867646\)
\(L(\frac12)\) \(\approx\) \(6.574867646\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
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

−13.97043358906687, −13.63667973080404, −13.14689114316789, −12.67527237540845, −11.95622925770598, −11.71960594585965, −11.09280755562849, −10.59970792590299, −10.12973207419958, −9.684059455402721, −8.771597812518727, −8.434278492704306, −8.061616007653127, −7.230192147009678, −6.572176443941464, −6.345489685802262, −5.676126902324592, −5.107386404610967, −4.697271204490248, −3.839649662895920, −3.496659158261299, −2.786589236652486, −2.069269226195511, −1.361932488100788, −0.8036202257214043, 0.8036202257214043, 1.361932488100788, 2.069269226195511, 2.786589236652486, 3.496659158261299, 3.839649662895920, 4.697271204490248, 5.107386404610967, 5.676126902324592, 6.345489685802262, 6.572176443941464, 7.230192147009678, 8.061616007653127, 8.434278492704306, 8.771597812518727, 9.684059455402721, 10.12973207419958, 10.59970792590299, 11.09280755562849, 11.71960594585965, 11.95622925770598, 12.67527237540845, 13.14689114316789, 13.63667973080404, 13.97043358906687

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