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Properties

Label 2-416-13.9-c3-0-0
Degree $2$
Conductor $416$
Sign $-0.779 + 0.626i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.49 + 4.32i)3-s + 1.37·5-s + (−3.31 − 5.75i)7-s + (1.05 + 1.82i)9-s + (7.64 − 13.2i)11-s + (−27.4 + 37.9i)13-s + (−3.41 + 5.92i)15-s + (39.7 + 68.8i)17-s + (−55.4 − 96.0i)19-s + 33.1·21-s + (−31.5 + 54.6i)23-s − 123.·25-s − 145.·27-s + (−61.0 + 105. i)29-s + 114.·31-s + ⋯
L(s)  = 1  + (−0.480 + 0.831i)3-s + 0.122·5-s + (−0.179 − 0.310i)7-s + (0.0391 + 0.0677i)9-s + (0.209 − 0.362i)11-s + (−0.585 + 0.810i)13-s + (−0.0588 + 0.101i)15-s + (0.566 + 0.981i)17-s + (−0.669 − 1.15i)19-s + 0.344·21-s + (−0.286 + 0.495i)23-s − 0.984·25-s − 1.03·27-s + (−0.390 + 0.677i)29-s + 0.661·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.779 + 0.626i$
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: \((2,\ 416,\ (\ :3/2),\ -0.779 + 0.626i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1387865732\)
\(L(\frac12)\) \(\approx\) \(0.1387865732\)
\(L(\frac{5}{2})\) 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 \( 1 + (27.4 - 37.9i)T \)
good3 \( 1 + (2.49 - 4.32i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 1.37T + 125T^{2} \)
7 \( 1 + (3.31 + 5.75i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-7.64 + 13.2i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-39.7 - 68.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (55.4 + 96.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (31.5 - 54.6i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (61.0 - 105. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 114.T + 2.97e4T^{2} \)
37 \( 1 + (-18.2 + 31.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-102. + 178. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (209. + 362. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 206.T + 1.03e5T^{2} \)
53 \( 1 + 346.T + 1.48e5T^{2} \)
59 \( 1 + (182. + 316. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (140. + 243. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (134. - 232. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (220. + 381. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 215.T + 3.89e5T^{2} \)
79 \( 1 + 604.T + 4.93e5T^{2} \)
83 \( 1 + 797.T + 5.71e5T^{2} \)
89 \( 1 + (-473. + 820. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-421. - 730. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.20102093085010714737396849116, −10.41149315500836234194878207917, −9.715213525938730067464495654357, −8.806694787907391668019388077470, −7.60627443390269763632578249650, −6.55240078926976038570051898062, −5.50508201824821653055120551223, −4.50719662759464326942161578170, −3.61750177540801562786272771194, −1.90288230572344353073856555776, 0.04799619714183880981876941104, 1.47802266991780870655857742780, 2.84250409004491551968917990565, 4.36051870454864634015462186915, 5.70679269530121385471277843267, 6.33865323802901304611807338155, 7.45515161359470179240537460184, 8.120983387570738725003741718331, 9.577656878105238644057628853057, 10.06699078657845253115274359091

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