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Properties

Label 2-230-5.3-c2-0-14
Degree $2$
Conductor $230$
Sign $0.258 + 0.965i$
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 − i)2-s + (1.74 + 1.74i)3-s − 2i·4-s + (−3.63 − 3.42i)5-s + 3.49·6-s + (4.68 − 4.68i)7-s + (−2 − 2i)8-s − 2.87i·9-s + (−7.06 + 0.211i)10-s + 7.34·11-s + (3.49 − 3.49i)12-s + (−0.802 − 0.802i)13-s − 9.37i·14-s + (−0.369 − 12.3i)15-s − 4·16-s + (2.03 − 2.03i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.583 + 0.583i)3-s − 0.5i·4-s + (−0.727 − 0.685i)5-s + 0.583·6-s + (0.669 − 0.669i)7-s + (−0.250 − 0.250i)8-s − 0.319i·9-s + (−0.706 + 0.0211i)10-s + 0.667·11-s + (0.291 − 0.291i)12-s + (−0.0617 − 0.0617i)13-s − 0.669i·14-s + (−0.0246 − 0.824i)15-s − 0.250·16-s + (0.119 − 0.119i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.258 + 0.965i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ 0.258 + 0.965i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.79790 - 1.37968i\)
\(L(\frac12)\) \(\approx\) \(1.79790 - 1.37968i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
5 \( 1 + (3.63 + 3.42i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good3 \( 1 + (-1.74 - 1.74i)T + 9iT^{2} \)
7 \( 1 + (-4.68 + 4.68i)T - 49iT^{2} \)
11 \( 1 - 7.34T + 121T^{2} \)
13 \( 1 + (0.802 + 0.802i)T + 169iT^{2} \)
17 \( 1 + (-2.03 + 2.03i)T - 289iT^{2} \)
19 \( 1 + 12.6iT - 361T^{2} \)
29 \( 1 - 18.2iT - 841T^{2} \)
31 \( 1 - 32.9T + 961T^{2} \)
37 \( 1 + (22.1 - 22.1i)T - 1.36e3iT^{2} \)
41 \( 1 + 16.8T + 1.68e3T^{2} \)
43 \( 1 + (-15.6 - 15.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (53.7 - 53.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (15.1 + 15.1i)T + 2.80e3iT^{2} \)
59 \( 1 + 8.32iT - 3.48e3T^{2} \)
61 \( 1 - 16.1T + 3.72e3T^{2} \)
67 \( 1 + (-21.8 + 21.8i)T - 4.48e3iT^{2} \)
71 \( 1 + 105.T + 5.04e3T^{2} \)
73 \( 1 + (-63.5 - 63.5i)T + 5.32e3iT^{2} \)
79 \( 1 - 19.7iT - 6.24e3T^{2} \)
83 \( 1 + (-105. - 105. i)T + 6.88e3iT^{2} \)
89 \( 1 - 131. iT - 7.92e3T^{2} \)
97 \( 1 + (-101. + 101. i)T - 9.40e3iT^{2} \)
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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

−11.76513486540493307558545303417, −11.00310700529841031807517532580, −9.822013153948283944444697861675, −8.958125351162413824005626197849, −8.010987648654716597536462088852, −6.69151437597372838801669560880, −4.94119485641247059748727206999, −4.21823278268419511293060893072, −3.21382166737967748148135215183, −1.11411374612332427703144571335, 2.12880348008222762983596237450, 3.47447776172304930171180492878, 4.80676650084662409616714429879, 6.20479373687271317225849514167, 7.25594559657186526685604731693, 8.060697971467841767024907215421, 8.756720080432885960382815734977, 10.37548848319745011517224875168, 11.60594221600406337673492168381, 12.11837790899454367884398202357

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