↑

Properties

Label 2-416-13.3-c3-0-6
Degree $2$
Conductor $416$
Sign $0.983 - 0.179i$
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  + (−5.10 − 8.84i)3-s + 2.27·5-s + (12.0 − 20.9i)7-s + (−38.6 + 66.8i)9-s + (16.8 + 29.1i)11-s + (46.6 + 4.13i)13-s + (−11.6 − 20.1i)15-s + (−47.5 + 82.2i)17-s + (−47.5 + 82.2i)19-s − 246.·21-s + (65.7 + 113. i)23-s − 119.·25-s + 512.·27-s + (134. + 233. i)29-s − 165.·31-s + ⋯
L(s)  = 1  + (−0.982 − 1.70i)3-s + 0.203·5-s + (0.651 − 1.12i)7-s + (−1.43 + 2.47i)9-s + (0.460 + 0.798i)11-s + (0.996 + 0.0881i)13-s + (−0.200 − 0.346i)15-s + (−0.677 + 1.17i)17-s + (−0.573 + 0.993i)19-s − 2.56·21-s + (0.595 + 1.03i)23-s − 0.958·25-s + 3.65·27-s + (0.863 + 1.49i)29-s − 0.958·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.983 - 0.179i$
Analytic conductor: \(24.5447\)
Root analytic conductor: \(4.95427\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :3/2),\ 0.983 - 0.179i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.047265496\)
\(L(\frac12)\) \(\approx\) \(1.047265496\)
\(L(\frac{5}{2})\) 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 \( 1 + (-46.6 - 4.13i)T \)
good3 \( 1 + (5.10 + 8.84i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 2.27T + 125T^{2} \)
7 \( 1 + (-12.0 + 20.9i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-16.8 - 29.1i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (47.5 - 82.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (47.5 - 82.2i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-65.7 - 113. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-134. - 233. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 165.T + 2.97e4T^{2} \)
37 \( 1 + (-7.07 - 12.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (46.5 + 80.7i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (55.3 - 95.9i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 133.T + 1.03e5T^{2} \)
53 \( 1 - 235.T + 1.48e5T^{2} \)
59 \( 1 + (-307. + 532. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (145. - 251. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-201. - 349. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-68.6 + 118. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 387.T + 3.89e5T^{2} \)
79 \( 1 + 608.T + 4.93e5T^{2} \)
83 \( 1 + 436.T + 5.71e5T^{2} \)
89 \( 1 + (-428. - 742. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (232. - 401. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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

−10.97873266189036437694700525350, −10.36152664385755656863983812369, −8.628116198188207911130831359138, −7.79253518433781244562024289360, −6.98688011572949978319156760750, −6.30921852530516002007336416491, −5.29707793537114039441936023049, −3.97349311873012322348642166235, −1.75027629263915210414010451882, −1.31988755232406571872055183494, 0.42657697318667608042210918112, 2.73741920266933868873032186841, 4.08740471986792967004515173387, 4.95043506813403817585444530226, 5.80173419602431458005811303245, 6.50133452505831651084024475125, 8.663152967921104827210175835937, 8.904162015116547507047121841470, 9.912869235251500105584907335216, 10.98856558760338924086219797863

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