Properties

Label 2-230-115.22-c3-0-5
Degree $2$
Conductor $230$
Sign $-0.909 - 0.415i$
Analytic cond. $13.5704$
Root an. cond. $3.68380$
Motivic weight $3$
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 − 1.41i)2-s + (−3.58 + 3.58i)3-s + 4.00i·4-s + (11.1 + 0.614i)5-s + 10.1·6-s + (−16.2 + 16.2i)7-s + (5.65 − 5.65i)8-s + 1.23i·9-s + (−14.9 − 16.6i)10-s + 34.5i·11-s + (−14.3 − 14.3i)12-s + (23.5 − 23.5i)13-s + 46.0·14-s + (−42.2 + 37.8i)15-s − 16.0·16-s + (55.6 − 55.6i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.690 + 0.690i)3-s + 0.500i·4-s + (0.998 + 0.0549i)5-s + 0.690·6-s + (−0.879 + 0.879i)7-s + (0.250 − 0.250i)8-s + 0.0457i·9-s + (−0.471 − 0.526i)10-s + 0.946i·11-s + (−0.345 − 0.345i)12-s + (0.503 − 0.503i)13-s + 0.879·14-s + (−0.727 + 0.651i)15-s − 0.250·16-s + (0.793 − 0.793i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.415i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.909 - 0.415i$
Analytic conductor: \(13.5704\)
Root analytic conductor: \(3.68380\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :3/2),\ -0.909 - 0.415i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.121748 + 0.560060i\)
\(L(\frac12)\) \(\approx\) \(0.121748 + 0.560060i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 1.41i)T \)
5 \( 1 + (-11.1 - 0.614i)T \)
23 \( 1 + (109. + 15.5i)T \)
good3 \( 1 + (3.58 - 3.58i)T - 27iT^{2} \)
7 \( 1 + (16.2 - 16.2i)T - 343iT^{2} \)
11 \( 1 - 34.5iT - 1.33e3T^{2} \)
13 \( 1 + (-23.5 + 23.5i)T - 2.19e3iT^{2} \)
17 \( 1 + (-55.6 + 55.6i)T - 4.91e3iT^{2} \)
19 \( 1 + 29.0T + 6.85e3T^{2} \)
29 \( 1 - 293. iT - 2.43e4T^{2} \)
31 \( 1 + 254.T + 2.97e4T^{2} \)
37 \( 1 + (82.6 - 82.6i)T - 5.06e4iT^{2} \)
41 \( 1 + 292.T + 6.89e4T^{2} \)
43 \( 1 + (295. + 295. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-277. - 277. i)T + 1.03e5iT^{2} \)
53 \( 1 + (185. + 185. i)T + 1.48e5iT^{2} \)
59 \( 1 - 26.0iT - 2.05e5T^{2} \)
61 \( 1 + 191. iT - 2.26e5T^{2} \)
67 \( 1 + (94.6 - 94.6i)T - 3.00e5iT^{2} \)
71 \( 1 - 17.9T + 3.57e5T^{2} \)
73 \( 1 + (272. - 272. i)T - 3.89e5iT^{2} \)
79 \( 1 - 1.03e3T + 4.93e5T^{2} \)
83 \( 1 + (819. + 819. i)T + 5.71e5iT^{2} \)
89 \( 1 + 394.T + 7.04e5T^{2} \)
97 \( 1 + (587. - 587. i)T - 9.12e5iT^{2} \)
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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

−12.18090177159331480169487840687, −10.93147097453644727735280039217, −10.13813212364229435769253831036, −9.614881955641535518825702072548, −8.657300313782081134205161868474, −7.07272431791395798726548677084, −5.83395188434145329854249707181, −5.03298895973538236147294357784, −3.29086587945099612451743389264, −1.91716819464745643933901795134, 0.29864042676744764806370009525, 1.58415229043518484790633963401, 3.72399145494608556877646091833, 5.75218922613385851576092872823, 6.19533231278389392388666835808, 7.03336712904512248353488502108, 8.301921425599600791094095758085, 9.485261168510082845958923319309, 10.24302635268283983018756362757, 11.19563534277431768079434256656

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