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Properties

Label 2-230-5.3-c4-0-9
Degree $2$
Conductor $230$
Sign $0.305 - 0.952i$
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 − 2i)2-s + (−3.68 − 3.68i)3-s − 8i·4-s + (6.02 + 24.2i)5-s − 14.7·6-s + (−4.81 + 4.81i)7-s + (−16 − 16i)8-s − 53.7i·9-s + (60.5 + 36.4i)10-s − 104.·11-s + (−29.5 + 29.5i)12-s + (89.3 + 89.3i)13-s + 19.2i·14-s + (67.2 − 111. i)15-s − 64·16-s + (−180. + 180. i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.409 − 0.409i)3-s − 0.5i·4-s + (0.241 + 0.970i)5-s − 0.409·6-s + (−0.0982 + 0.0982i)7-s + (−0.250 − 0.250i)8-s − 0.663i·9-s + (0.605 + 0.364i)10-s − 0.860·11-s + (−0.204 + 0.204i)12-s + (0.528 + 0.528i)13-s + 0.0982i·14-s + (0.298 − 0.496i)15-s − 0.250·16-s + (−0.625 + 0.625i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.305 - 0.952i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ 0.305 - 0.952i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.178215563\)
\(L(\frac12)\) \(\approx\) \(1.178215563\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 + 2i)T \)
5 \( 1 + (-6.02 - 24.2i)T \)
23 \( 1 + (77.9 + 77.9i)T \)
good3 \( 1 + (3.68 + 3.68i)T + 81iT^{2} \)
7 \( 1 + (4.81 - 4.81i)T - 2.40e3iT^{2} \)
11 \( 1 + 104.T + 1.46e4T^{2} \)
13 \( 1 + (-89.3 - 89.3i)T + 2.85e4iT^{2} \)
17 \( 1 + (180. - 180. i)T - 8.35e4iT^{2} \)
19 \( 1 - 475. iT - 1.30e5T^{2} \)
29 \( 1 - 506. iT - 7.07e5T^{2} \)
31 \( 1 - 387.T + 9.23e5T^{2} \)
37 \( 1 + (1.01e3 - 1.01e3i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.89e3T + 2.82e6T^{2} \)
43 \( 1 + (-1.14e3 - 1.14e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-1.19e3 + 1.19e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-1.96e3 - 1.96e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 22.0iT - 1.21e7T^{2} \)
61 \( 1 + 4.64e3T + 1.38e7T^{2} \)
67 \( 1 + (2.93e3 - 2.93e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 5.44e3T + 2.54e7T^{2} \)
73 \( 1 + (200. + 200. i)T + 2.83e7iT^{2} \)
79 \( 1 + 6.69e3iT - 3.89e7T^{2} \)
83 \( 1 + (4.55e3 + 4.55e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.21e3iT - 6.27e7T^{2} \)
97 \( 1 + (-5.31e3 + 5.31e3i)T - 8.85e7iT^{2} \)
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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

−11.79592883135225723173939559763, −10.82984738400867654060381926913, −10.19784331591801902701520629157, −8.961718485357188775335737171053, −7.50180874231003285243442801740, −6.35115063374554390947813990365, −5.80625420532822843255453588352, −4.10749411506210806382622758994, −2.92806112047841643994968894736, −1.55747104848854710757265106398, 0.34360730186364327114617943917, 2.48054006436761404260550406605, 4.26808832200248323675988832112, 5.10660868827684850748182123837, 5.84461007104108532874637828030, 7.31077056957257052283438776049, 8.316923988931871362352334146914, 9.294898575830065981948793524131, 10.50327733526263218578395674746, 11.36000322854927626595274331876

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