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Properties

Label 2-392-8.3-c2-0-68
Degree $2$
Conductor $392$
Sign $0.156 + 0.987i$
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  + (1.67 − 1.08i)2-s + 3.98·3-s + (1.62 − 3.65i)4-s − 1.88i·5-s + (6.67 − 4.33i)6-s + (−1.25 − 7.90i)8-s + 6.84·9-s + (−2.05 − 3.15i)10-s − 7.87·11-s + (6.47 − 14.5i)12-s − 11.4i·13-s − 7.49i·15-s + (−10.7 − 11.8i)16-s + 2.89·17-s + (11.4 − 7.46i)18-s + 30.0·19-s + ⋯
L(s)  = 1  + (0.838 − 0.544i)2-s + 1.32·3-s + (0.406 − 0.913i)4-s − 0.376i·5-s + (1.11 − 0.722i)6-s + (−0.156 − 0.987i)8-s + 0.760·9-s + (−0.205 − 0.315i)10-s − 0.716·11-s + (0.539 − 1.21i)12-s − 0.883i·13-s − 0.499i·15-s + (−0.669 − 0.742i)16-s + 0.170·17-s + (0.638 − 0.414i)18-s + 1.58·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.156 + 0.987i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.156 + 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.06567 - 2.61783i\)
\(L(\frac12)\) \(\approx\) \(3.06567 - 2.61783i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.67 + 1.08i)T \)
7 \( 1 \)
good3 \( 1 - 3.98T + 9T^{2} \)
5 \( 1 + 1.88iT - 25T^{2} \)
11 \( 1 + 7.87T + 121T^{2} \)
13 \( 1 + 11.4iT - 169T^{2} \)
17 \( 1 - 2.89T + 289T^{2} \)
19 \( 1 - 30.0T + 361T^{2} \)
23 \( 1 - 38.5iT - 529T^{2} \)
29 \( 1 - 27.8iT - 841T^{2} \)
31 \( 1 + 22.4iT - 961T^{2} \)
37 \( 1 - 45.5iT - 1.36e3T^{2} \)
41 \( 1 - 40.6T + 1.68e3T^{2} \)
43 \( 1 + 47.2T + 1.84e3T^{2} \)
47 \( 1 - 82.5iT - 2.20e3T^{2} \)
53 \( 1 + 26.8iT - 2.80e3T^{2} \)
59 \( 1 - 10.4T + 3.48e3T^{2} \)
61 \( 1 + 22.1iT - 3.72e3T^{2} \)
67 \( 1 + 59.3T + 4.48e3T^{2} \)
71 \( 1 + 38.2iT - 5.04e3T^{2} \)
73 \( 1 + 13.9T + 5.32e3T^{2} \)
79 \( 1 + 51.1iT - 6.24e3T^{2} \)
83 \( 1 - 89.4T + 6.88e3T^{2} \)
89 \( 1 + 105.T + 7.92e3T^{2} \)
97 \( 1 + 55.3T + 9.40e3T^{2} \)
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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

−10.95887993368304814407915959648, −9.860926663264530346639747420768, −9.285396311410239863502812545670, −8.054438993075303047157718846407, −7.32652574571303403400039930437, −5.71246636865030342331648540082, −4.88335170178286321903384518116, −3.39267366147270769198004434375, −2.89003366026984647715326256370, −1.35288633934123065448894512469, 2.33282517740839710012628281949, 3.12663130883399074849761287973, 4.22378748431409830910172134117, 5.41679401735961754682084116612, 6.73984382816359442409674946732, 7.51192559345918766482330987602, 8.371574885799693485742452988627, 9.158755032722107530706883301599, 10.35835283815799680404465075981, 11.49266843120484758098635829042

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