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Properties

Label 2-392-56.45-c2-0-39
Degree $2$
Conductor $392$
Sign $0.378 + 0.925i$
Analytic cond. $10.6812$
Root an. cond. $3.26821$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.70 − 1.04i)2-s + (−2.78 + 4.81i)3-s + (1.83 + 3.55i)4-s + (1.52 + 2.64i)5-s + (9.76 − 5.32i)6-s + (0.579 − 7.97i)8-s + (−10.9 − 18.9i)9-s + (0.147 − 6.11i)10-s + (0.106 + 0.0612i)11-s + (−22.2 − 1.07i)12-s − 4.11·13-s − 17.0·15-s + (−9.30 + 13.0i)16-s + (−17.8 − 10.3i)17-s + (−1.05 + 43.8i)18-s + (−4.46 − 7.74i)19-s + ⋯
L(s)  = 1  + (−0.853 − 0.520i)2-s + (−0.926 + 1.60i)3-s + (0.457 + 0.889i)4-s + (0.305 + 0.529i)5-s + (1.62 − 0.887i)6-s + (0.0724 − 0.997i)8-s + (−1.21 − 2.10i)9-s + (0.0147 − 0.611i)10-s + (0.00963 + 0.00556i)11-s + (−1.85 − 0.0895i)12-s − 0.316·13-s − 1.13·15-s + (−0.581 + 0.813i)16-s + (−1.05 − 0.606i)17-s + (−0.0588 + 2.43i)18-s + (−0.235 − 0.407i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.378 + 0.925i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.378 + 0.925i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.236914 - 0.159157i\)
\(L(\frac12)\) \(\approx\) \(0.236914 - 0.159157i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.70 + 1.04i)T \)
7 \( 1 \)
good3 \( 1 + (2.78 - 4.81i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-1.52 - 2.64i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (-0.106 - 0.0612i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 4.11T + 169T^{2} \)
17 \( 1 + (17.8 + 10.3i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (4.46 + 7.74i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-7.51 - 13.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 31.6iT - 841T^{2} \)
31 \( 1 + (23.0 + 13.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (25.1 - 14.5i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 9.26iT - 1.68e3T^{2} \)
43 \( 1 - 45.3iT - 1.84e3T^{2} \)
47 \( 1 + (-68.6 + 39.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (55.0 + 31.7i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-14.2 + 24.6i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-12.6 - 21.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-65.4 - 37.8i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 2.81T + 5.04e3T^{2} \)
73 \( 1 + (11.0 + 6.40i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (35.6 + 61.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 30.0T + 6.88e3T^{2} \)
89 \( 1 + (15.3 - 8.83i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 26.1iT - 9.40e3T^{2} \)
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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

−10.89716047423389321678031333120, −10.01823257713206484943890915907, −9.476778223717707588438235017314, −8.626766111993483614469387364622, −7.08170296789412447908121123267, −6.14488561292387562313955920457, −4.85164879323642612647044934952, −3.83849666541573512224209083135, −2.59370288399317874552892157033, −0.20285886133959872715431681642, 1.19590671803786040621434814432, 2.17732047680760179774950267495, 5.00986490113740888652365216521, 5.81549726573362483710364740280, 6.72309999749508008668213901543, 7.33229504113264907341176399466, 8.391348839460551398607201728300, 9.100803020466399709998709586475, 10.62790591184368167869712056443, 11.06539884935560163801641980306

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