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Properties

Label 2-176-11.4-c3-0-12
Degree $2$
Conductor $176$
Sign $0.490 + 0.871i$
Analytic cond. $10.3843$
Root an. cond. $3.22247$
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  + (−2.97 + 9.14i)3-s + (13.5 − 9.82i)5-s + (−6.15 − 18.9i)7-s + (−52.8 − 38.4i)9-s + (−17.1 − 32.2i)11-s + (−32.0 − 23.2i)13-s + (49.6 + 152. i)15-s + (1.66 − 1.21i)17-s + (31.3 − 96.4i)19-s + 191.·21-s − 110.·23-s + (47.7 − 146. i)25-s + (298. − 216. i)27-s + (−10.3 − 31.7i)29-s + (140. + 102. i)31-s + ⋯
L(s)  = 1  + (−0.571 + 1.75i)3-s + (1.20 − 0.878i)5-s + (−0.332 − 1.02i)7-s + (−1.95 − 1.42i)9-s + (−0.470 − 0.882i)11-s + (−0.683 − 0.496i)13-s + (0.854 + 2.62i)15-s + (0.0237 − 0.0172i)17-s + (0.378 − 1.16i)19-s + 1.98·21-s − 1.00·23-s + (0.381 − 1.17i)25-s + (2.12 − 1.54i)27-s + (−0.0660 − 0.203i)29-s + (0.814 + 0.591i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.490 + 0.871i$
Analytic conductor: \(10.3843\)
Root analytic conductor: \(3.22247\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :3/2),\ 0.490 + 0.871i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.904801 - 0.528720i\)
\(L(\frac12)\) \(\approx\) \(0.904801 - 0.528720i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (17.1 + 32.2i)T \)
good3 \( 1 + (2.97 - 9.14i)T + (-21.8 - 15.8i)T^{2} \)
5 \( 1 + (-13.5 + 9.82i)T + (38.6 - 118. i)T^{2} \)
7 \( 1 + (6.15 + 18.9i)T + (-277. + 201. i)T^{2} \)
13 \( 1 + (32.0 + 23.2i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (-1.66 + 1.21i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (-31.3 + 96.4i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 + 110.T + 1.21e4T^{2} \)
29 \( 1 + (10.3 + 31.7i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (-140. - 102. i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (-77.6 - 239. i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (59.5 - 183. i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 + 304.T + 7.95e4T^{2} \)
47 \( 1 + (-10.5 + 32.4i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (175. + 127. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (-4.96 - 15.2i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (-429. + 312. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 + 636.T + 3.00e5T^{2} \)
71 \( 1 + (-227. + 165. i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (202. + 624. i)T + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (-652. - 473. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (-516. + 375. i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 - 1.14e3T + 7.04e5T^{2} \)
97 \( 1 + (679. + 493. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−11.85339238485171769157562846335, −10.72753268706213257926115059850, −10.00788104275642552242145934328, −9.504439011733952415925016497404, −8.345310965184241847263349933523, −6.35315818962101808498696513088, −5.29515790705547921475695407407, −4.62044949120761637638337777063, −3.15315562785271811322050836101, −0.48211533375631553394247737783, 1.90291472654983951010368085768, 2.49802703411646590584199988323, 5.47197762124456495600207366888, 6.13822456243583401628053455510, 6.99513957028602411789480183532, 7.970742497729579019910019603663, 9.500989307722976059342556781691, 10.45417285755641464298666564193, 11.86385124298826430190076538593, 12.32388066973210007255481524639

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