Properties

Label 2-234-1.1-c7-0-0
Degree $2$
Conductor $234$
Sign $1$
Analytic cond. $73.0980$
Root an. cond. $8.54974$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 64·4-s − 321·5-s − 181·7-s − 512·8-s + 2.56e3·10-s − 7.78e3·11-s + 2.19e3·13-s + 1.44e3·14-s + 4.09e3·16-s − 9.06e3·17-s − 3.71e4·19-s − 2.05e4·20-s + 6.22e4·22-s − 1.90e4·23-s + 2.49e4·25-s − 1.75e4·26-s − 1.15e4·28-s − 1.74e5·29-s + 2.90e4·31-s − 3.27e4·32-s + 7.25e4·34-s + 5.81e4·35-s + 3.23e5·37-s + 2.97e5·38-s + 1.64e5·40-s − 7.95e5·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.14·5-s − 0.199·7-s − 0.353·8-s + 0.812·10-s − 1.76·11-s + 0.277·13-s + 0.141·14-s + 1/4·16-s − 0.447·17-s − 1.24·19-s − 0.574·20-s + 1.24·22-s − 0.325·23-s + 0.318·25-s − 0.196·26-s − 0.0997·28-s − 1.33·29-s + 0.174·31-s − 0.176·32-s + 0.316·34-s + 0.229·35-s + 1.05·37-s + 0.878·38-s + 0.406·40-s − 1.80·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(73.0980\)
Root analytic conductor: \(8.54974\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.2130683452\)
\(L(\frac12)\) \(\approx\) \(0.2130683452\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{3} T \)
3 \( 1 \)
13 \( 1 - p^{3} T \)
good5 \( 1 + 321 T + p^{7} T^{2} \)
7 \( 1 + 181 T + p^{7} T^{2} \)
11 \( 1 + 7782 T + p^{7} T^{2} \)
17 \( 1 + 9069 T + p^{7} T^{2} \)
19 \( 1 + 37150 T + p^{7} T^{2} \)
23 \( 1 + 19008 T + p^{7} T^{2} \)
29 \( 1 + 174750 T + p^{7} T^{2} \)
31 \( 1 - 29012 T + p^{7} T^{2} \)
37 \( 1 - 323669 T + p^{7} T^{2} \)
41 \( 1 + 795312 T + p^{7} T^{2} \)
43 \( 1 + 314137 T + p^{7} T^{2} \)
47 \( 1 - 447441 T + p^{7} T^{2} \)
53 \( 1 - 1469232 T + p^{7} T^{2} \)
59 \( 1 + 1627770 T + p^{7} T^{2} \)
61 \( 1 + 2399608 T + p^{7} T^{2} \)
67 \( 1 + 64066 T + p^{7} T^{2} \)
71 \( 1 - 322383 T + p^{7} T^{2} \)
73 \( 1 + 4454782 T + p^{7} T^{2} \)
79 \( 1 - 753560 T + p^{7} T^{2} \)
83 \( 1 - 1219092 T + p^{7} T^{2} \)
89 \( 1 + 3390330 T + p^{7} T^{2} \)
97 \( 1 - 1628774 T + p^{7} T^{2} \)
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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

−10.84374934796994507024563368206, −10.08556727655082096544377456359, −8.741600544041272077941958995829, −8.007533850636215702079373717890, −7.27302447314138650371684567458, −5.98732688655820463205045579301, −4.59299214038010309193046536438, −3.32043381477138303491128461878, −2.07342175925158254397547004051, −0.24813169619351851849933699289, 0.24813169619351851849933699289, 2.07342175925158254397547004051, 3.32043381477138303491128461878, 4.59299214038010309193046536438, 5.98732688655820463205045579301, 7.27302447314138650371684567458, 8.007533850636215702079373717890, 8.741600544041272077941958995829, 10.08556727655082096544377456359, 10.84374934796994507024563368206

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