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Properties

Label 2-416-13.9-c1-0-1
Degree $2$
Conductor $416$
Sign $-0.0128 - 0.999i$
Analytic cond. $3.32177$
Root an. cond. $1.82257$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.207 − 0.358i)3-s − 2.82·5-s + (0.792 + 1.37i)7-s + (1.41 + 2.44i)9-s + (−2.62 + 4.54i)11-s + (−1 − 3.46i)13-s + (−0.585 + 1.01i)15-s + (0.0857 + 0.148i)17-s + (3.62 + 6.27i)19-s + 0.656·21-s + (−3.62 + 6.27i)23-s + 3.00·25-s + 2.41·27-s + (1.32 − 2.30i)29-s + 5.65·31-s + ⋯
L(s)  = 1  + (0.119 − 0.207i)3-s − 1.26·5-s + (0.299 + 0.519i)7-s + (0.471 + 0.816i)9-s + (−0.790 + 1.36i)11-s + (−0.277 − 0.960i)13-s + (−0.151 + 0.261i)15-s + (0.0208 + 0.0360i)17-s + (0.830 + 1.43i)19-s + 0.143·21-s + (−0.755 + 1.30i)23-s + 0.600·25-s + 0.464·27-s + (0.246 − 0.427i)29-s + 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.0128 - 0.999i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.649541 + 0.657924i\)
\(L(\frac12)\) \(\approx\) \(0.649541 + 0.657924i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (1 + 3.46i)T \)
good3 \( 1 + (-0.207 + 0.358i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 2.82T + 5T^{2} \)
7 \( 1 + (-0.792 - 1.37i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.62 - 4.54i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.0857 - 0.148i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.62 - 6.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.62 - 6.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.32 + 2.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + (4.74 - 8.21i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0857 + 0.148i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.03 + 8.72i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 - 2.82T + 53T^{2} \)
59 \( 1 + (-3.62 - 6.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.37 + 4.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.621 - 1.07i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.48T + 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-7.32 + 12.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.5 - 7.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.83669854895216553203583891629, −10.32361202741907574121250582253, −9.950637336760670557166965531338, −8.196595566786757182725774877896, −7.890454310843357315933530154568, −7.16301982154919587188416679669, −5.48198528747405048092641515389, −4.66746644552499408139395255332, −3.39697408338966575749145793737, −1.92262402817580827586058454024, 0.59675671118218918460236937320, 2.97781434624446805719630255174, 4.02518619012058607441005479869, 4.85611813310988074459383364679, 6.44681822157245728736762298299, 7.32545419090055812618387222217, 8.242032035378495833313999001254, 9.020135412983367824107771367280, 10.18257573336889572028925933437, 11.14705572895428063448877417336

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