Properties

Label 2-230-23.22-c2-0-15
Degree $2$
Conductor $230$
Sign $0.653 + 0.756i$
Analytic cond. $6.26704$
Root an. cond. $2.50340$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + 1.43·3-s + 2.00·4-s − 2.23i·5-s + 2.03·6-s − 10.1i·7-s + 2.82·8-s − 6.93·9-s − 3.16i·10-s − 13.0i·11-s + 2.87·12-s + 23.2·13-s − 14.4i·14-s − 3.21i·15-s + 4.00·16-s + 28.2i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.479·3-s + 0.500·4-s − 0.447i·5-s + 0.339·6-s − 1.45i·7-s + 0.353·8-s − 0.770·9-s − 0.316i·10-s − 1.18i·11-s + 0.239·12-s + 1.78·13-s − 1.02i·14-s − 0.214i·15-s + 0.250·16-s + 1.66i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.756i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.653 + 0.756i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ 0.653 + 0.756i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.43502 - 1.11376i\)
\(L(\frac12)\) \(\approx\) \(2.43502 - 1.11376i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
5 \( 1 + 2.23iT \)
23 \( 1 + (17.3 - 15.0i)T \)
good3 \( 1 - 1.43T + 9T^{2} \)
7 \( 1 + 10.1iT - 49T^{2} \)
11 \( 1 + 13.0iT - 121T^{2} \)
13 \( 1 - 23.2T + 169T^{2} \)
17 \( 1 - 28.2iT - 289T^{2} \)
19 \( 1 - 11.6iT - 361T^{2} \)
29 \( 1 - 42.4T + 841T^{2} \)
31 \( 1 - 18.7T + 961T^{2} \)
37 \( 1 - 1.14iT - 1.36e3T^{2} \)
41 \( 1 + 72.8T + 1.68e3T^{2} \)
43 \( 1 - 4.96iT - 1.84e3T^{2} \)
47 \( 1 + 0.813T + 2.20e3T^{2} \)
53 \( 1 + 26.7iT - 2.80e3T^{2} \)
59 \( 1 - 94.0T + 3.48e3T^{2} \)
61 \( 1 - 74.5iT - 3.72e3T^{2} \)
67 \( 1 + 80.0iT - 4.48e3T^{2} \)
71 \( 1 + 83.5T + 5.04e3T^{2} \)
73 \( 1 + 8.98T + 5.32e3T^{2} \)
79 \( 1 - 80.2iT - 6.24e3T^{2} \)
83 \( 1 - 94.6iT - 6.88e3T^{2} \)
89 \( 1 - 136. iT - 7.92e3T^{2} \)
97 \( 1 - 2.32iT - 9.40e3T^{2} \)
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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

−11.84749236509897044628502727162, −10.93244005268378726604430520954, −10.21125018685290030263036431357, −8.367122512689244958946541652077, −8.264630350181920620896956393389, −6.49440339297643145684994836184, −5.71336164799353883823464019089, −3.98741456362268215446203569484, −3.45069279993252010704355840805, −1.27586641431590126930787814755, 2.29264505855285128930203618846, 3.15341868216811998201284442434, 4.76733668683285657280481742888, 5.90761910066620950344980482915, 6.83570676272023224198915788654, 8.285139031527876226081664011528, 9.019707283255461302075919399430, 10.24010233197330744404772012364, 11.63247036121648038370055364422, 11.88160744099560859767059054504

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