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Properties

Label 2-230-23.22-c4-0-29
Degree $2$
Conductor $230$
Sign $0.0129 + 0.999i$
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  − 2.82·2-s + 15.6·3-s + 8.00·4-s − 11.1i·5-s − 44.3·6-s − 65.6i·7-s − 22.6·8-s + 165.·9-s + 31.6i·10-s − 29.7i·11-s + 125.·12-s − 101.·13-s + 185. i·14-s − 175. i·15-s + 64.0·16-s − 114. i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.74·3-s + 0.500·4-s − 0.447i·5-s − 1.23·6-s − 1.33i·7-s − 0.353·8-s + 2.03·9-s + 0.316i·10-s − 0.245i·11-s + 0.871·12-s − 0.601·13-s + 0.946i·14-s − 0.779i·15-s + 0.250·16-s − 0.394i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0129 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0129 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.0129 + 0.999i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ 0.0129 + 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.320033979\)
\(L(\frac12)\) \(\approx\) \(2.320033979\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82T \)
5 \( 1 + 11.1iT \)
23 \( 1 + (528. - 6.85i)T \)
good3 \( 1 - 15.6T + 81T^{2} \)
7 \( 1 + 65.6iT - 2.40e3T^{2} \)
11 \( 1 + 29.7iT - 1.46e4T^{2} \)
13 \( 1 + 101.T + 2.85e4T^{2} \)
17 \( 1 + 114. iT - 8.35e4T^{2} \)
19 \( 1 + 373. iT - 1.30e5T^{2} \)
29 \( 1 + 512.T + 7.07e5T^{2} \)
31 \( 1 - 883.T + 9.23e5T^{2} \)
37 \( 1 - 811. iT - 1.87e6T^{2} \)
41 \( 1 + 397.T + 2.82e6T^{2} \)
43 \( 1 - 2.00e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.91e3T + 4.87e6T^{2} \)
53 \( 1 + 1.56e3iT - 7.89e6T^{2} \)
59 \( 1 - 521.T + 1.21e7T^{2} \)
61 \( 1 + 3.79e3iT - 1.38e7T^{2} \)
67 \( 1 + 6.22e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.63e3T + 2.54e7T^{2} \)
73 \( 1 - 7.63e3T + 2.83e7T^{2} \)
79 \( 1 + 8.97e3iT - 3.89e7T^{2} \)
83 \( 1 - 8.22e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.15e4iT - 6.27e7T^{2} \)
97 \( 1 - 7.77e3iT - 8.85e7T^{2} \)
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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

−11.00532037392604416542900592454, −9.890252784684284602631061590246, −9.383121370697713129076148110673, −8.280653737298587477920830446861, −7.67242659069977634847571514417, −6.77548631330854601826950043658, −4.60356295160058212983559469782, −3.46228353064187971373431475140, −2.20625755968965611154364622511, −0.77236052320445382708190900846, 1.94616501417738286417605908535, 2.60576902116085067588183492711, 3.87275524315432181898863415063, 5.81193019386857162224781663228, 7.21044280644236993846105302969, 8.091570363127679097701698637659, 8.782167041340828545913744054074, 9.632927396393343975337578684635, 10.34417751500992553725868286855, 11.91628541149822556282230991275

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