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Properties

Label 2-230-5.4-c5-0-36
Degree $2$
Conductor $230$
Sign $-0.591 + 0.806i$
Analytic cond. $36.8882$
Root an. cond. $6.07357$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·2-s − 17.9i·3-s − 16·4-s + (−33.0 + 45.0i)5-s − 71.8·6-s + 13.4i·7-s + 64i·8-s − 79.6·9-s + (180. + 132. i)10-s + 554.·11-s + 287. i·12-s − 117. i·13-s + 53.7·14-s + (809. + 593. i)15-s + 256·16-s + 288. i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.15i·3-s − 0.5·4-s + (−0.591 + 0.806i)5-s − 0.814·6-s + 0.103i·7-s + 0.353i·8-s − 0.327·9-s + (0.570 + 0.418i)10-s + 1.38·11-s + 0.576i·12-s − 0.193i·13-s + 0.0733·14-s + (0.929 + 0.681i)15-s + 0.250·16-s + 0.241i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 + 0.806i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.591 + 0.806i$
Analytic conductor: \(36.8882\)
Root analytic conductor: \(6.07357\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :5/2),\ -0.591 + 0.806i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.737248509\)
\(L(\frac12)\) \(\approx\) \(1.737248509\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4iT \)
5 \( 1 + (33.0 - 45.0i)T \)
23 \( 1 + 529iT \)
good3 \( 1 + 17.9iT - 243T^{2} \)
7 \( 1 - 13.4iT - 1.68e4T^{2} \)
11 \( 1 - 554.T + 1.61e5T^{2} \)
13 \( 1 + 117. iT - 3.71e5T^{2} \)
17 \( 1 - 288. iT - 1.41e6T^{2} \)
19 \( 1 - 515.T + 2.47e6T^{2} \)
29 \( 1 - 8.06e3T + 2.05e7T^{2} \)
31 \( 1 + 6.69e3T + 2.86e7T^{2} \)
37 \( 1 + 3.91e3iT - 6.93e7T^{2} \)
41 \( 1 - 4.43e3T + 1.15e8T^{2} \)
43 \( 1 - 1.43e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.12e4iT - 2.29e8T^{2} \)
53 \( 1 - 5.76e3iT - 4.18e8T^{2} \)
59 \( 1 + 4.86e4T + 7.14e8T^{2} \)
61 \( 1 + 2.62e3T + 8.44e8T^{2} \)
67 \( 1 + 1.38e3iT - 1.35e9T^{2} \)
71 \( 1 - 4.01e4T + 1.80e9T^{2} \)
73 \( 1 + 3.01e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.32e4T + 3.07e9T^{2} \)
83 \( 1 + 6.90e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.00e5T + 5.58e9T^{2} \)
97 \( 1 + 1.04e5iT - 8.58e9T^{2} \)
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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

−11.23766826796358182689916128166, −10.24242241325077742075725515551, −9.017728513667361059707678056593, −7.922913115933012713304382493413, −6.98829642788986334192480491047, −6.14271721028837305616383321688, −4.30323121633910994335840587811, −3.14133530064241892400024813202, −1.84299728405157180409265115087, −0.65638294714361405319324287324, 1.05661005182092074882194792494, 3.61719992554863920062805372796, 4.36520853953752022404482272622, 5.23161588927212839077376871248, 6.61183460937121617733711114315, 7.77287853527045094645092495833, 9.021634481652572426208532566907, 9.331747436272562217127839191325, 10.55896926291234802545441449496, 11.70469578553259903255975848095

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