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Properties

Label 2-392-56.51-c2-0-23
Degree $2$
Conductor $392$
Sign $0.678 - 0.734i$
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  + (−0.944 − 1.76i)2-s + (1.72 + 2.98i)3-s + (−2.21 + 3.32i)4-s + (4.22 + 2.44i)5-s + (3.62 − 5.84i)6-s + (7.96 + 0.763i)8-s + (−1.42 + 2.46i)9-s + (0.311 − 9.76i)10-s + (10.7 + 18.6i)11-s + (−13.7 − 0.876i)12-s − 13.0i·13-s + 16.8i·15-s + (−6.17 − 14.7i)16-s + (0.117 + 0.203i)17-s + (5.68 + 0.181i)18-s + (−2.27 + 3.94i)19-s + ⋯
L(s)  = 1  + (−0.472 − 0.881i)2-s + (0.573 + 0.993i)3-s + (−0.554 + 0.832i)4-s + (0.845 + 0.488i)5-s + (0.604 − 0.974i)6-s + (0.995 + 0.0954i)8-s + (−0.157 + 0.273i)9-s + (0.0311 − 0.976i)10-s + (0.976 + 1.69i)11-s + (−1.14 − 0.0730i)12-s − 1.00i·13-s + 1.12i·15-s + (−0.385 − 0.922i)16-s + (0.00690 + 0.0119i)17-s + (0.315 + 0.0100i)18-s + (−0.119 + 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.678 - 0.734i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.67358 + 0.732221i\)
\(L(\frac12)\) \(\approx\) \(1.67358 + 0.732221i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.944 + 1.76i)T \)
7 \( 1 \)
good3 \( 1 + (-1.72 - 2.98i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-4.22 - 2.44i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-10.7 - 18.6i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 13.0iT - 169T^{2} \)
17 \( 1 + (-0.117 - 0.203i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (2.27 - 3.94i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (9.48 + 5.47i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 34.6iT - 841T^{2} \)
31 \( 1 + (29.5 - 17.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-46.9 - 27.1i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 37.8T + 1.68e3T^{2} \)
43 \( 1 + 4.84T + 1.84e3T^{2} \)
47 \( 1 + (-62.6 - 36.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-18.7 + 10.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (17.4 + 30.2i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (55.0 + 31.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (9.21 + 15.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 47.5iT - 5.04e3T^{2} \)
73 \( 1 + (27.9 + 48.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-82.2 - 47.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 71.5T + 6.88e3T^{2} \)
89 \( 1 + (-79.8 + 138. i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 90.4T + 9.40e3T^{2} \)
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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

−10.73111812382435092380573917743, −10.20807286906550553614476454954, −9.561306177046170632902164576083, −8.976245973996006635799815430112, −7.74492141810139727603448410413, −6.59304996542249303890052360585, −4.95090993738154230446878766259, −3.97171314248688768787486882882, −2.92194377473301583444797347387, −1.70400351589242813481818546470, 0.968974615728403693347946329459, 2.06717299969662254840978413541, 4.07402110401641055242406940086, 5.65084461305233886055387015396, 6.28262450447498755040785118479, 7.24309273786522517540403940439, 8.237802906170041263667284470316, 8.981328862993978659179501253152, 9.527739201699992027998437126138, 10.88313707404880030527405378812

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