Properties

Label 2-230-115.22-c3-0-7
Degree $2$
Conductor $230$
Sign $0.193 - 0.981i$
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 + (3.25 − 3.25i)3-s + 4.00i·4-s + (7.62 − 8.17i)5-s − 9.21·6-s + (−22.5 + 22.5i)7-s + (5.65 − 5.65i)8-s + 5.75i·9-s + (−22.3 + 0.787i)10-s + 20.5i·11-s + (13.0 + 13.0i)12-s + (−37.5 + 37.5i)13-s + 63.7·14-s + (−1.81 − 51.4i)15-s − 16.0·16-s + (−96.6 + 96.6i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.627 − 0.627i)3-s + 0.500i·4-s + (0.681 − 0.731i)5-s − 0.627·6-s + (−1.21 + 1.21i)7-s + (0.250 − 0.250i)8-s + 0.213i·9-s + (−0.706 + 0.0249i)10-s + 0.562i·11-s + (0.313 + 0.313i)12-s + (−0.800 + 0.800i)13-s + 1.21·14-s + (−0.0312 − 0.886i)15-s − 0.250·16-s + (−1.37 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.193 - 0.981i$
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: \((2,\ 230,\ (\ :3/2),\ 0.193 - 0.981i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.592428 + 0.486991i\)
\(L(\frac12)\) \(\approx\) \(0.592428 + 0.486991i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-7.62 + 8.17i)T \)
23 \( 1 + (-77.8 + 78.1i)T \)
good3 \( 1 + (-3.25 + 3.25i)T - 27iT^{2} \)
7 \( 1 + (22.5 - 22.5i)T - 343iT^{2} \)
11 \( 1 - 20.5iT - 1.33e3T^{2} \)
13 \( 1 + (37.5 - 37.5i)T - 2.19e3iT^{2} \)
17 \( 1 + (96.6 - 96.6i)T - 4.91e3iT^{2} \)
19 \( 1 + 51.6T + 6.85e3T^{2} \)
29 \( 1 + 204. iT - 2.43e4T^{2} \)
31 \( 1 + 328.T + 2.97e4T^{2} \)
37 \( 1 + (86.2 - 86.2i)T - 5.06e4iT^{2} \)
41 \( 1 - 97.1T + 6.89e4T^{2} \)
43 \( 1 + (-268. - 268. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-191. - 191. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-256. - 256. i)T + 1.48e5iT^{2} \)
59 \( 1 - 20.3iT - 2.05e5T^{2} \)
61 \( 1 + 350. iT - 2.26e5T^{2} \)
67 \( 1 + (80.2 - 80.2i)T - 3.00e5iT^{2} \)
71 \( 1 - 711.T + 3.57e5T^{2} \)
73 \( 1 + (159. - 159. i)T - 3.89e5iT^{2} \)
79 \( 1 + 31.7T + 4.93e5T^{2} \)
83 \( 1 + (421. + 421. i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.52e3T + 7.04e5T^{2} \)
97 \( 1 + (-601. + 601. i)T - 9.12e5iT^{2} \)
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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

−12.46965618331581869519180589544, −10.92240089497703200986727591690, −9.698134718281134546288731665419, −9.047559101051628047559065387149, −8.438180659011820361231447043727, −7.00908521325979353160166872755, −6.00085738956004363701959831513, −4.41453668938835144402789363893, −2.47263810697596718458340096216, −1.98186555810906943775219645958, 0.31219265530972012407465946465, 2.74261600653160610278309928598, 3.77618339305567569077452145186, 5.45157649873219895109727412696, 6.86945361111650636299704587320, 7.19294381545914465322657666247, 9.003947527374230489099690480098, 9.446429347827778233043119422513, 10.39788326906244842587297716647, 10.95182130787766232448995830948

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