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Properties

Label 2-230-5.3-c4-0-26
Degree $2$
Conductor $230$
Sign $0.638 + 0.769i$
Analytic cond. $23.7750$
Root an. cond. $4.87597$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 2i)2-s + (1.43 + 1.43i)3-s − 8i·4-s + (23.6 + 7.95i)5-s + 5.74·6-s + (−17.1 + 17.1i)7-s + (−16 − 16i)8-s − 76.8i·9-s + (63.3 − 31.4i)10-s + 18.4·11-s + (11.4 − 11.4i)12-s + (107. + 107. i)13-s + 68.6i·14-s + (22.6 + 45.4i)15-s − 64·16-s + (341. − 341. i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.159 + 0.159i)3-s − 0.5i·4-s + (0.947 + 0.318i)5-s + 0.159·6-s + (−0.350 + 0.350i)7-s + (−0.250 − 0.250i)8-s − 0.949i·9-s + (0.633 − 0.314i)10-s + 0.152·11-s + (0.0798 − 0.0798i)12-s + (0.637 + 0.637i)13-s + 0.350i·14-s + (0.100 + 0.202i)15-s − 0.250·16-s + (1.18 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.638 + 0.769i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ 0.638 + 0.769i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.213146159\)
\(L(\frac12)\) \(\approx\) \(3.213146159\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 + 2i)T \)
5 \( 1 + (-23.6 - 7.95i)T \)
23 \( 1 + (-77.9 - 77.9i)T \)
good3 \( 1 + (-1.43 - 1.43i)T + 81iT^{2} \)
7 \( 1 + (17.1 - 17.1i)T - 2.40e3iT^{2} \)
11 \( 1 - 18.4T + 1.46e4T^{2} \)
13 \( 1 + (-107. - 107. i)T + 2.85e4iT^{2} \)
17 \( 1 + (-341. + 341. i)T - 8.35e4iT^{2} \)
19 \( 1 + 229. iT - 1.30e5T^{2} \)
29 \( 1 - 1.14e3iT - 7.07e5T^{2} \)
31 \( 1 - 663.T + 9.23e5T^{2} \)
37 \( 1 + (-1.51e3 + 1.51e3i)T - 1.87e6iT^{2} \)
41 \( 1 - 599.T + 2.82e6T^{2} \)
43 \( 1 + (2.32e3 + 2.32e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-1.33e3 + 1.33e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-521. - 521. i)T + 7.89e6iT^{2} \)
59 \( 1 + 5.21e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.30e3T + 1.38e7T^{2} \)
67 \( 1 + (4.57e3 - 4.57e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 2.60e3T + 2.54e7T^{2} \)
73 \( 1 + (5.62e3 + 5.62e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 2.13e3iT - 3.89e7T^{2} \)
83 \( 1 + (-9.14e3 - 9.14e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 3.19e3iT - 6.27e7T^{2} \)
97 \( 1 + (6.19e3 - 6.19e3i)T - 8.85e7iT^{2} \)
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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

−11.53270116207112554339629603773, −10.42784654208902226550476461981, −9.447509756031111620915174389862, −8.998263341504865653588480701234, −7.04426272426357632488537187127, −6.16010755024860251713142293359, −5.14677657263933757328354933006, −3.60690665121087000577615319908, −2.63293160918938316402467159198, −1.07241691556249056521282054367, 1.37880308623975792934751773650, 2.94305807490681558993357214545, 4.38379113132618708360787995633, 5.67796607393149152392532132832, 6.29730069885433436285791704696, 7.78339420563310690957297621562, 8.414682323943551217288893966275, 9.871194534044075432191614161224, 10.51306917461027533426231493314, 11.92734585555199668409165046827

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