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Properties

Label 2-416-13.9-c3-0-33
Degree $2$
Conductor $416$
Sign $0.275 + 0.961i$
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  + (1.10 − 1.91i)3-s + 15.5·5-s + (−13.4 − 23.2i)7-s + (11.0 + 19.1i)9-s + (24.1 − 41.7i)11-s + (24.3 + 40.0i)13-s + (17.2 − 29.8i)15-s + (−54.6 − 94.7i)17-s + (27.4 + 47.5i)19-s − 59.4·21-s + (33.6 − 58.2i)23-s + 117.·25-s + 108.·27-s + (5.91 − 10.2i)29-s − 194.·31-s + ⋯
L(s)  = 1  + (0.212 − 0.368i)3-s + 1.39·5-s + (−0.726 − 1.25i)7-s + (0.409 + 0.709i)9-s + (0.660 − 1.14i)11-s + (0.520 + 0.853i)13-s + (0.296 − 0.512i)15-s + (−0.780 − 1.35i)17-s + (0.331 + 0.573i)19-s − 0.617·21-s + (0.304 − 0.527i)23-s + 0.938·25-s + 0.773·27-s + (0.0378 − 0.0655i)29-s − 1.12·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.275 + 0.961i$
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: \((2,\ 416,\ (\ :3/2),\ 0.275 + 0.961i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.577568991\)
\(L(\frac12)\) \(\approx\) \(2.577568991\)
\(L(\frac{5}{2})\) 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 \( 1 + (-24.3 - 40.0i)T \)
good3 \( 1 + (-1.10 + 1.91i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 15.5T + 125T^{2} \)
7 \( 1 + (13.4 + 23.2i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-24.1 + 41.7i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (54.6 + 94.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-27.4 - 47.5i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-33.6 + 58.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-5.91 + 10.2i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 194.T + 2.97e4T^{2} \)
37 \( 1 + (-118. + 205. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (69.2 - 119. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-148. - 257. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 261.T + 1.03e5T^{2} \)
53 \( 1 + 229.T + 1.48e5T^{2} \)
59 \( 1 + (114. + 197. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (319. + 552. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-329. + 570. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (509. + 882. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 1.05e3T + 3.89e5T^{2} \)
79 \( 1 - 452.T + 4.93e5T^{2} \)
83 \( 1 + 369.T + 5.71e5T^{2} \)
89 \( 1 + (-432. + 749. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-576. - 999. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.64836788802920094005866451662, −9.544955749836807569470832422419, −9.089306508849715988754736413942, −7.67154532310715652739042192622, −6.71543443994214838305541406728, −6.11995586522290632567111614786, −4.73611336167966508556595642946, −3.46137344646375495820010374697, −2.06721017526770228047251292487, −0.873209591203661017980281401939, 1.56241275394630151639373705060, 2.69161674009912694134012702943, 3.97196706522896622784662101576, 5.43008821039443618808006394089, 6.14716469095208819342632795126, 6.96619619372606198916855404713, 8.700763814882221282336058919780, 9.276752092217597835572867895528, 9.829927643995174252831048901260, 10.70548880747184789814682763141

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