Properties

Label 2-230-115.64-c1-0-7
Degree $2$
Conductor $230$
Sign $0.0320 + 0.999i$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.281 − 0.959i)2-s + (−0.951 − 0.824i)3-s + (−0.841 − 0.540i)4-s + (1.50 + 1.65i)5-s + (−1.05 + 0.680i)6-s + (4.71 − 2.15i)7-s + (−0.755 + 0.654i)8-s + (−0.201 − 1.40i)9-s + (2.01 − 0.973i)10-s + (−2.19 + 0.643i)11-s + (0.354 + 1.20i)12-s + (−2.34 − 1.06i)13-s + (−0.737 − 5.13i)14-s + (−0.0625 − 2.81i)15-s + (0.415 + 0.909i)16-s + (−3.46 − 5.39i)17-s + ⋯
L(s)  = 1  + (0.199 − 0.678i)2-s + (−0.549 − 0.475i)3-s + (−0.420 − 0.270i)4-s + (0.671 + 0.741i)5-s + (−0.432 + 0.277i)6-s + (1.78 − 0.814i)7-s + (−0.267 + 0.231i)8-s + (−0.0671 − 0.467i)9-s + (0.636 − 0.307i)10-s + (−0.660 + 0.193i)11-s + (0.102 + 0.348i)12-s + (−0.649 − 0.296i)13-s + (−0.197 − 1.37i)14-s + (−0.0161 − 0.726i)15-s + (0.103 + 0.227i)16-s + (−0.840 − 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0320 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0320 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.0320 + 0.999i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.0320 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.951070 - 0.921016i\)
\(L(\frac12)\) \(\approx\) \(0.951070 - 0.921016i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.281 + 0.959i)T \)
5 \( 1 + (-1.50 - 1.65i)T \)
23 \( 1 + (-4.75 + 0.619i)T \)
good3 \( 1 + (0.951 + 0.824i)T + (0.426 + 2.96i)T^{2} \)
7 \( 1 + (-4.71 + 2.15i)T + (4.58 - 5.29i)T^{2} \)
11 \( 1 + (2.19 - 0.643i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (2.34 + 1.06i)T + (8.51 + 9.82i)T^{2} \)
17 \( 1 + (3.46 + 5.39i)T + (-7.06 + 15.4i)T^{2} \)
19 \( 1 + (-5.04 - 3.24i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (1.24 - 0.797i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (-0.0386 - 0.0446i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (-2.82 + 0.405i)T + (35.5 - 10.4i)T^{2} \)
41 \( 1 + (1.69 - 11.7i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-2.17 - 1.88i)T + (6.11 + 42.5i)T^{2} \)
47 \( 1 + 2.51iT - 47T^{2} \)
53 \( 1 + (8.37 - 3.82i)T + (34.7 - 40.0i)T^{2} \)
59 \( 1 + (4.59 - 10.0i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (1.22 + 1.41i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (1.16 - 3.96i)T + (-56.3 - 36.2i)T^{2} \)
71 \( 1 + (-2.50 - 0.736i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (3.81 - 5.94i)T + (-30.3 - 66.4i)T^{2} \)
79 \( 1 + (2.52 - 5.52i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (0.603 - 0.0868i)T + (79.6 - 23.3i)T^{2} \)
89 \( 1 + (-1.26 + 1.45i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (5.40 + 0.776i)T + (93.0 + 27.3i)T^{2} \)
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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

−11.61484731354034632998186048361, −11.26294456391495112443689316370, −10.33107267792529443712127942538, −9.347795126436518550269798385667, −7.75495089218392316322675463316, −7.00940857590169636083308150752, −5.51656152556894989580487102544, −4.68494470767423143901148153195, −2.84441576961020435081129588889, −1.32402066087190433694474186531, 2.08423565666869929264948472523, 4.69091845637884149013326410682, 5.07097829395020365786585983452, 5.88704954533885757030961547820, 7.59384708082053400798715115266, 8.479733959405438784418014750327, 9.288392044668524167256321520097, 10.66782968608478996201312315315, 11.42471281120446011467790518691, 12.48014285859965217841329756044

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