↑

Properties

Label 2-476-119.100-c1-0-9
Degree $2$
Conductor $476$
Sign $0.223 + 0.974i$
Analytic cond. $3.80087$
Root an. cond. $1.94958$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.886 − 1.15i)3-s + (0.518 − 3.94i)5-s + (2.61 + 0.415i)7-s + (0.227 + 0.847i)9-s + (2.28 − 0.300i)11-s + 4.52i·13-s + (−4.09 − 4.09i)15-s + (−0.547 − 4.08i)17-s + (−2.87 + 0.770i)19-s + (2.79 − 2.65i)21-s + (−2.82 − 3.68i)23-s + (−10.4 − 2.79i)25-s + (5.21 + 2.16i)27-s + (−4.64 + 1.92i)29-s + (0.254 − 0.331i)31-s + ⋯
L(s)  = 1  + (0.512 − 0.667i)3-s + (0.231 − 1.76i)5-s + (0.987 + 0.156i)7-s + (0.0757 + 0.282i)9-s + (0.687 − 0.0905i)11-s + 1.25i·13-s + (−1.05 − 1.05i)15-s + (−0.132 − 0.991i)17-s + (−0.659 + 0.176i)19-s + (0.610 − 0.578i)21-s + (−0.589 − 0.768i)23-s + (−2.08 − 0.558i)25-s + (1.00 + 0.416i)27-s + (−0.862 + 0.357i)29-s + (0.0457 − 0.0595i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.223 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.223 + 0.974i$
Analytic conductor: \(3.80087\)
Root analytic conductor: \(1.94958\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 476,\ (\ :1/2),\ 0.223 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49213 - 1.18926i\)
\(L(\frac12)\) \(\approx\) \(1.49213 - 1.18926i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.61 - 0.415i)T \)
17 \( 1 + (0.547 + 4.08i)T \)
good3 \( 1 + (-0.886 + 1.15i)T + (-0.776 - 2.89i)T^{2} \)
5 \( 1 + (-0.518 + 3.94i)T + (-4.82 - 1.29i)T^{2} \)
11 \( 1 + (-2.28 + 0.300i)T + (10.6 - 2.84i)T^{2} \)
13 \( 1 - 4.52iT - 13T^{2} \)
19 \( 1 + (2.87 - 0.770i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.82 + 3.68i)T + (-5.95 + 22.2i)T^{2} \)
29 \( 1 + (4.64 - 1.92i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-0.254 + 0.331i)T + (-8.02 - 29.9i)T^{2} \)
37 \( 1 + (0.943 + 0.124i)T + (35.7 + 9.57i)T^{2} \)
41 \( 1 + (-8.74 - 3.62i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (5.20 - 5.20i)T - 43iT^{2} \)
47 \( 1 + (0.958 + 0.553i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.02 - 11.3i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.85 - 0.496i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.73 - 2.10i)T + (15.7 - 58.9i)T^{2} \)
67 \( 1 + (3.32 + 5.76i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.42 + 10.6i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-12.0 - 9.25i)T + (18.8 + 70.5i)T^{2} \)
79 \( 1 + (-2.39 - 3.12i)T + (-20.4 + 76.3i)T^{2} \)
83 \( 1 + (8.54 + 8.54i)T + 83iT^{2} \)
89 \( 1 + (1.85 + 1.07i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.41 + 2.65i)T + (68.5 - 68.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.02340807815594098961505973364, −9.502064819030904113588969855110, −8.892688871377735605201923131532, −8.255286033767270152480442283720, −7.39155746865926792936067602224, −6.11888890240369802458808967240, −4.84046447645341237696647842450, −4.34107940739383624690187729147, −2.13630809956384951098598621173, −1.32724325613033089129413517808, 2.08003116017442547286756795267, 3.38475732666248270031149857692, 4.06742042795531453649709436396, 5.65762706273498288262689197401, 6.60663358294945280362799740875, 7.59542089466918676768691559882, 8.466037611227604754830450485006, 9.635462253922987523133540301400, 10.37778152556075513499402974878, 10.92796376161286743479839306822

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