Properties

Label 2-230-23.12-c3-0-9
Degree $2$
Conductor $230$
Sign $0.997 - 0.0661i$
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  + (−1.68 + 1.08i)2-s + (−1.10 + 7.65i)3-s + (1.66 − 3.63i)4-s + (−4.79 + 1.40i)5-s + (−6.42 − 14.0i)6-s + (−11.9 − 13.7i)7-s + (1.13 + 7.91i)8-s + (−31.4 − 9.22i)9-s + (6.54 − 7.55i)10-s + (10.1 + 6.52i)11-s + (26.0 + 16.7i)12-s + (59.7 − 68.9i)13-s + (34.9 + 10.2i)14-s + (−5.50 − 38.2i)15-s + (−10.4 − 12.0i)16-s + (−23.5 − 51.4i)17-s + ⋯
L(s)  = 1  + (−0.594 + 0.382i)2-s + (−0.211 + 1.47i)3-s + (0.207 − 0.454i)4-s + (−0.429 + 0.125i)5-s + (−0.436 − 0.956i)6-s + (−0.644 − 0.743i)7-s + (0.0503 + 0.349i)8-s + (−1.16 − 0.341i)9-s + (0.207 − 0.238i)10-s + (0.278 + 0.178i)11-s + (0.625 + 0.402i)12-s + (1.27 − 1.47i)13-s + (0.667 + 0.195i)14-s + (−0.0946 − 0.658i)15-s + (−0.163 − 0.188i)16-s + (−0.335 − 0.734i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0661i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.997 - 0.0661i$
Analytic conductor: \(13.5704\)
Root analytic conductor: \(3.68380\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :3/2),\ 0.997 - 0.0661i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.799828 + 0.0265028i\)
\(L(\frac12)\) \(\approx\) \(0.799828 + 0.0265028i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.68 - 1.08i)T \)
5 \( 1 + (4.79 - 1.40i)T \)
23 \( 1 + (80.7 + 75.1i)T \)
good3 \( 1 + (1.10 - 7.65i)T + (-25.9 - 7.60i)T^{2} \)
7 \( 1 + (11.9 + 13.7i)T + (-48.8 + 339. i)T^{2} \)
11 \( 1 + (-10.1 - 6.52i)T + (552. + 1.21e3i)T^{2} \)
13 \( 1 + (-59.7 + 68.9i)T + (-312. - 2.17e3i)T^{2} \)
17 \( 1 + (23.5 + 51.4i)T + (-3.21e3 + 3.71e3i)T^{2} \)
19 \( 1 + (18.4 - 40.3i)T + (-4.49e3 - 5.18e3i)T^{2} \)
29 \( 1 + (-109. - 240. i)T + (-1.59e4 + 1.84e4i)T^{2} \)
31 \( 1 + (20.9 + 145. i)T + (-2.85e4 + 8.39e3i)T^{2} \)
37 \( 1 + (-184. - 54.2i)T + (4.26e4 + 2.73e4i)T^{2} \)
41 \( 1 + (-217. + 63.8i)T + (5.79e4 - 3.72e4i)T^{2} \)
43 \( 1 + (-55.5 + 386. i)T + (-7.62e4 - 2.23e4i)T^{2} \)
47 \( 1 + 352.T + 1.03e5T^{2} \)
53 \( 1 + (-101. - 117. i)T + (-2.11e4 + 1.47e5i)T^{2} \)
59 \( 1 + (-51.7 + 59.7i)T + (-2.92e4 - 2.03e5i)T^{2} \)
61 \( 1 + (-48.2 - 335. i)T + (-2.17e5 + 6.39e4i)T^{2} \)
67 \( 1 + (-614. + 394. i)T + (1.24e5 - 2.73e5i)T^{2} \)
71 \( 1 + (-383. + 246. i)T + (1.48e5 - 3.25e5i)T^{2} \)
73 \( 1 + (-443. + 970. i)T + (-2.54e5 - 2.93e5i)T^{2} \)
79 \( 1 + (696. - 803. i)T + (-7.01e4 - 4.88e5i)T^{2} \)
83 \( 1 + (-220. - 64.6i)T + (4.81e5 + 3.09e5i)T^{2} \)
89 \( 1 + (-182. + 1.26e3i)T + (-6.76e5 - 1.98e5i)T^{2} \)
97 \( 1 + (640. - 188. i)T + (7.67e5 - 4.93e5i)T^{2} \)
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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

−11.22755215341623714429100314947, −10.54795026523285871103579504270, −9.995068499869858532352539831134, −8.958762796948780163670305916319, −7.977279484806482223030576839816, −6.68107034794265271999323020834, −5.55594464602661490455209568217, −4.24000682866299447057282031181, −3.29156315935002374169605900378, −0.48643645531642948174804051685, 1.15053843246339495207088502433, 2.34369335453401671818677408199, 3.97466368242454381812300615965, 6.19494783129150349696328404975, 6.57791787352644089379686435524, 7.912179629222691012856819837370, 8.664208931945159453237069743882, 9.607732714448629053772803659787, 11.28848938704920799377227201689, 11.62499306460675064675961202338

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