Properties

Label 2-230-115.22-c3-0-18
Degree $2$
Conductor $230$
Sign $0.776 - 0.630i$
Analytic cond. $13.5704$
Root an. cond. $3.68380$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 − 1.41i)2-s + (−7.12 + 7.12i)3-s + 4.00i·4-s + (1.63 + 11.0i)5-s + 20.1·6-s + (16.1 − 16.1i)7-s + (5.65 − 5.65i)8-s − 74.4i·9-s + (13.3 − 17.9i)10-s + 27.4i·11-s + (−28.4 − 28.4i)12-s + (38.2 − 38.2i)13-s − 45.5·14-s + (−90.3 − 67.1i)15-s − 16.0·16-s + (24.6 − 24.6i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−1.37 + 1.37i)3-s + 0.500i·4-s + (0.145 + 0.989i)5-s + 1.37·6-s + (0.870 − 0.870i)7-s + (0.250 − 0.250i)8-s − 2.75i·9-s + (0.421 − 0.567i)10-s + 0.752i·11-s + (−0.685 − 0.685i)12-s + (0.815 − 0.815i)13-s − 0.870·14-s + (−1.55 − 1.15i)15-s − 0.250·16-s + (0.352 − 0.352i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 - 0.630i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.776 - 0.630i$
Analytic conductor: \(13.5704\)
Root analytic conductor: \(3.68380\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :3/2),\ 0.776 - 0.630i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.930784 + 0.330516i\)
\(L(\frac12)\) \(\approx\) \(0.930784 + 0.330516i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 1.41i)T \)
5 \( 1 + (-1.63 - 11.0i)T \)
23 \( 1 + (76.6 + 79.3i)T \)
good3 \( 1 + (7.12 - 7.12i)T - 27iT^{2} \)
7 \( 1 + (-16.1 + 16.1i)T - 343iT^{2} \)
11 \( 1 - 27.4iT - 1.33e3T^{2} \)
13 \( 1 + (-38.2 + 38.2i)T - 2.19e3iT^{2} \)
17 \( 1 + (-24.6 + 24.6i)T - 4.91e3iT^{2} \)
19 \( 1 - 142.T + 6.85e3T^{2} \)
29 \( 1 - 43.7iT - 2.43e4T^{2} \)
31 \( 1 - 206.T + 2.97e4T^{2} \)
37 \( 1 + (-62.6 + 62.6i)T - 5.06e4iT^{2} \)
41 \( 1 - 104.T + 6.89e4T^{2} \)
43 \( 1 + (6.53 + 6.53i)T + 7.95e4iT^{2} \)
47 \( 1 + (123. + 123. i)T + 1.03e5iT^{2} \)
53 \( 1 + (414. + 414. i)T + 1.48e5iT^{2} \)
59 \( 1 - 636. iT - 2.05e5T^{2} \)
61 \( 1 - 411. iT - 2.26e5T^{2} \)
67 \( 1 + (-154. + 154. i)T - 3.00e5iT^{2} \)
71 \( 1 - 615.T + 3.57e5T^{2} \)
73 \( 1 + (-222. + 222. i)T - 3.89e5iT^{2} \)
79 \( 1 - 322.T + 4.93e5T^{2} \)
83 \( 1 + (-940. - 940. i)T + 5.71e5iT^{2} \)
89 \( 1 + 834.T + 7.04e5T^{2} \)
97 \( 1 + (-629. + 629. i)T - 9.12e5iT^{2} \)
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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

−11.44203615025971463474266038820, −10.80134971395225721423133385466, −10.17584583839170118581448660828, −9.582828020292076687936470274411, −7.891180433473648481987080282948, −6.73478124043346671541447573289, −5.48499387944851129228471012442, −4.36539322094026664509072385239, −3.30436200907392005189315824029, −0.885348308514358524407421947688, 0.924110978842420325039033789977, 1.74178904101981622986662164038, 4.93033735189315910751676659946, 5.69478299944416863313773476403, 6.35175854068359895045406513161, 7.82824155580036694946215693265, 8.261526898605533973110291772740, 9.542825263927296998264381093182, 11.12731178982565847332803283181, 11.67328699626581294018870248167

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