Properties

Label 2-230-23.22-c2-0-5
Degree $2$
Conductor $230$
Sign $-0.254 + 0.967i$
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 − 5.41·3-s + 2.00·4-s − 2.23i·5-s + 7.66·6-s + 8.24i·7-s − 2.82·8-s + 20.3·9-s + 3.16i·10-s + 15.8i·11-s − 10.8·12-s − 14.3·13-s − 11.6i·14-s + 12.1i·15-s + 4.00·16-s − 10.1i·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.80·3-s + 0.500·4-s − 0.447i·5-s + 1.27·6-s + 1.17i·7-s − 0.353·8-s + 2.26·9-s + 0.316i·10-s + 1.43i·11-s − 0.903·12-s − 1.10·13-s − 0.832i·14-s + 0.807i·15-s + 0.250·16-s − 0.598i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.254 + 0.967i$
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: \((2,\ 230,\ (\ :1),\ -0.254 + 0.967i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.147545 - 0.191337i\)
\(L(\frac12)\) \(\approx\) \(0.147545 - 0.191337i\)
\(L(2)\) 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 \( 1 + (-22.2 - 5.84i)T \)
good3 \( 1 + 5.41T + 9T^{2} \)
7 \( 1 - 8.24iT - 49T^{2} \)
11 \( 1 - 15.8iT - 121T^{2} \)
13 \( 1 + 14.3T + 169T^{2} \)
17 \( 1 + 10.1iT - 289T^{2} \)
19 \( 1 + 36.5iT - 361T^{2} \)
29 \( 1 - 6.46T + 841T^{2} \)
31 \( 1 + 42.8T + 961T^{2} \)
37 \( 1 + 63.6iT - 1.36e3T^{2} \)
41 \( 1 + 37.0T + 1.68e3T^{2} \)
43 \( 1 - 6.00iT - 1.84e3T^{2} \)
47 \( 1 + 32.4T + 2.20e3T^{2} \)
53 \( 1 + 36.6iT - 2.80e3T^{2} \)
59 \( 1 - 6.65T + 3.48e3T^{2} \)
61 \( 1 + 55.7iT - 3.72e3T^{2} \)
67 \( 1 - 4.45iT - 4.48e3T^{2} \)
71 \( 1 - 118.T + 5.04e3T^{2} \)
73 \( 1 - 82.2T + 5.32e3T^{2} \)
79 \( 1 + 133. iT - 6.24e3T^{2} \)
83 \( 1 + 67.5iT - 6.88e3T^{2} \)
89 \( 1 - 104. iT - 7.92e3T^{2} \)
97 \( 1 + 98.6iT - 9.40e3T^{2} \)
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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

−11.63853423388903362563214539813, −10.91135040749510304398437959967, −9.667379533366011273644813280258, −9.170199438784641790292162797851, −7.35252435503514435310127118808, −6.74739336182501520206121853151, −5.27130891075658291907434332896, −4.89362661694179991162623717106, −2.12504045406519273001646631012, −0.23738347240363406452066612275, 1.16022260764845045917918850914, 3.71160434675757575011063568814, 5.25443053973467670280118081200, 6.28638391102918740656331753402, 7.06087316028652854249305902183, 8.096445166921115076403163349273, 9.864662745591925308718345913450, 10.50724161776968928408523932955, 11.05145855662610522961346565798, 11.92505901501058652698454317732

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