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Properties

Label 2-230-115.114-c4-0-45
Degree $2$
Conductor $230$
Sign $-0.979 + 0.203i$
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  + 2.82i·2-s − 15.5i·3-s − 8.00·4-s + (22.6 − 10.6i)5-s + 43.9·6-s − 25.0·7-s − 22.6i·8-s − 160.·9-s + (30.1 + 63.9i)10-s − 156. i·11-s + 124. i·12-s − 61.5i·13-s − 70.8i·14-s + (−165. − 351. i)15-s + 64.0·16-s + 38.3·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.72i·3-s − 0.500·4-s + (0.904 − 0.426i)5-s + 1.22·6-s − 0.511·7-s − 0.353i·8-s − 1.98·9-s + (0.301 + 0.639i)10-s − 1.29i·11-s + 0.863i·12-s − 0.364i·13-s − 0.361i·14-s + (−0.736 − 1.56i)15-s + 0.250·16-s + 0.132·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.979 + 0.203i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ -0.979 + 0.203i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.166121816\)
\(L(\frac12)\) \(\approx\) \(1.166121816\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
5 \( 1 + (-22.6 + 10.6i)T \)
23 \( 1 + (123. - 514. i)T \)
good3 \( 1 + 15.5iT - 81T^{2} \)
7 \( 1 + 25.0T + 2.40e3T^{2} \)
11 \( 1 + 156. iT - 1.46e4T^{2} \)
13 \( 1 + 61.5iT - 2.85e4T^{2} \)
17 \( 1 - 38.3T + 8.35e4T^{2} \)
19 \( 1 - 19.8iT - 1.30e5T^{2} \)
29 \( 1 + 1.10e3T + 7.07e5T^{2} \)
31 \( 1 - 1.30e3T + 9.23e5T^{2} \)
37 \( 1 + 2.52e3T + 1.87e6T^{2} \)
41 \( 1 - 1.79e3T + 2.82e6T^{2} \)
43 \( 1 + 3.19e3T + 3.41e6T^{2} \)
47 \( 1 - 1.73e3iT - 4.87e6T^{2} \)
53 \( 1 + 98.3T + 7.89e6T^{2} \)
59 \( 1 - 2.86e3T + 1.21e7T^{2} \)
61 \( 1 + 2.29e3iT - 1.38e7T^{2} \)
67 \( 1 - 4.97e3T + 2.01e7T^{2} \)
71 \( 1 + 3.01e3T + 2.54e7T^{2} \)
73 \( 1 - 2.80e3iT - 2.83e7T^{2} \)
79 \( 1 + 6.13e3iT - 3.89e7T^{2} \)
83 \( 1 + 5.32e3T + 4.74e7T^{2} \)
89 \( 1 + 5.44e3iT - 6.27e7T^{2} \)
97 \( 1 + 9.24e3T + 8.85e7T^{2} \)
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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

−11.34873083162856535265424618773, −9.873554846731657098067940031274, −8.734850814466646645890256343877, −8.023606134424303811757749824035, −6.90427599089315931805832303564, −6.07109129624493681002145638266, −5.44226649697984702900671661339, −3.15250330052420020939819005379, −1.58580581207597682135933649197, −0.38152185727123139783541060588, 2.16759655755659036105209016780, 3.38211120601105727956953457725, 4.48249451591740954065918505351, 5.41284153134469696419236498975, 6.76039833307992579125711960512, 8.642478892316688985641259888826, 9.631678738530185904520560632600, 9.981594466113760137442926131655, 10.67499088837824720025508622514, 11.73633998576988824166881279231

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