↑

Properties

Label 2-416-13.9-c3-0-40
Degree $2$
Conductor $416$
Sign $-0.830 - 0.557i$
Analytic cond. $24.5447$
Root an. cond. $4.95427$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.36 − 7.55i)3-s − 11.4·5-s + (−4.58 − 7.93i)7-s + (−24.5 − 42.5i)9-s + (31.5 − 54.5i)11-s + (−44.4 − 14.8i)13-s + (−49.7 + 86.2i)15-s + (11.8 + 20.4i)17-s + (51.1 + 88.6i)19-s − 79.9·21-s + (−4.51 + 7.81i)23-s + 5.27·25-s − 192.·27-s + (−73.5 + 127. i)29-s − 91.4·31-s + ⋯
L(s)  = 1  + (0.839 − 1.45i)3-s − 1.02·5-s + (−0.247 − 0.428i)7-s + (−0.909 − 1.57i)9-s + (0.863 − 1.49i)11-s + (−0.948 − 0.317i)13-s + (−0.856 + 1.48i)15-s + (0.168 + 0.292i)17-s + (0.617 + 1.07i)19-s − 0.830·21-s + (−0.0409 + 0.0708i)23-s + 0.0422·25-s − 1.37·27-s + (−0.470 + 0.815i)29-s − 0.530·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.830 - 0.557i$
Analytic conductor: \(24.5447\)
Root analytic conductor: \(4.95427\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :3/2),\ -0.830 - 0.557i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.094005947\)
\(L(\frac12)\) \(\approx\) \(1.094005947\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (44.4 + 14.8i)T \)
good3 \( 1 + (-4.36 + 7.55i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 11.4T + 125T^{2} \)
7 \( 1 + (4.58 + 7.93i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-31.5 + 54.5i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-11.8 - 20.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-51.1 - 88.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (4.51 - 7.81i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (73.5 - 127. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 91.4T + 2.97e4T^{2} \)
37 \( 1 + (-69.1 + 119. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (183. - 317. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (191. + 332. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 529.T + 1.03e5T^{2} \)
53 \( 1 - 458.T + 1.48e5T^{2} \)
59 \( 1 + (-339. - 587. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-143. - 248. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-47.5 + 82.3i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (433. + 751. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 695.T + 3.89e5T^{2} \)
79 \( 1 - 518.T + 4.93e5T^{2} \)
83 \( 1 + 764.T + 5.71e5T^{2} \)
89 \( 1 + (-396. + 685. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-243. - 421. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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

−10.23913858523577106592933037805, −8.955317550777383939455758302458, −8.216666294099041283423829411062, −7.53244450416319687837913648356, −6.81934939141469594401791011064, −5.69124191780442912242994964076, −3.77040459635380588527185508482, −3.15004253319277661801523665721, −1.51335069181788638495966083062, −0.32378839805104207237662974319, 2.34241704698771717983679087125, 3.51733309661857700433956471639, 4.39042427067565074234089195165, 5.05892057710795326215373509977, 6.91019047143627145439248535286, 7.76237272175869091686582326649, 8.860613755026163011296517940781, 9.599333349653054849903079563877, 10.00863844230270810360006828029, 11.44415949498658702603123299230

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