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Properties

Label 2-392-56.13-c2-0-63
Degree $2$
Conductor $392$
Sign $-0.995 + 0.0907i$
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  + (−0.621 − 1.90i)2-s + 3.40·3-s + (−3.22 + 2.36i)4-s − 4.31·5-s + (−2.11 − 6.46i)6-s + (6.49 + 4.66i)8-s + 2.57·9-s + (2.68 + 8.20i)10-s − 17.8i·11-s + (−10.9 + 8.04i)12-s + 3.25·13-s − 14.6·15-s + (4.83 − 15.2i)16-s − 15.7i·17-s + (−1.60 − 4.90i)18-s − 1.55·19-s + ⋯
L(s)  = 1  + (−0.310 − 0.950i)2-s + 1.13·3-s + (−0.806 + 0.590i)4-s − 0.863·5-s + (−0.352 − 1.07i)6-s + (0.812 + 0.583i)8-s + 0.286·9-s + (0.268 + 0.820i)10-s − 1.62i·11-s + (−0.915 + 0.670i)12-s + 0.250·13-s − 0.979·15-s + (0.302 − 0.953i)16-s − 0.925i·17-s + (−0.0890 − 0.272i)18-s − 0.0819·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0907i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.995 + 0.0907i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ -0.995 + 0.0907i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0445169 - 0.979553i\)
\(L(\frac12)\) \(\approx\) \(0.0445169 - 0.979553i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.621 + 1.90i)T \)
7 \( 1 \)
good3 \( 1 - 3.40T + 9T^{2} \)
5 \( 1 + 4.31T + 25T^{2} \)
11 \( 1 + 17.8iT - 121T^{2} \)
13 \( 1 - 3.25T + 169T^{2} \)
17 \( 1 + 15.7iT - 289T^{2} \)
19 \( 1 + 1.55T + 361T^{2} \)
23 \( 1 + 41.4T + 529T^{2} \)
29 \( 1 - 3.74iT - 841T^{2} \)
31 \( 1 + 0.0167iT - 961T^{2} \)
37 \( 1 - 1.34iT - 1.36e3T^{2} \)
41 \( 1 + 70.3iT - 1.68e3T^{2} \)
43 \( 1 + 13.0iT - 1.84e3T^{2} \)
47 \( 1 + 35.7iT - 2.20e3T^{2} \)
53 \( 1 - 45.9iT - 2.80e3T^{2} \)
59 \( 1 - 68.7T + 3.48e3T^{2} \)
61 \( 1 - 96.0T + 3.72e3T^{2} \)
67 \( 1 - 13.9iT - 4.48e3T^{2} \)
71 \( 1 + 75.7T + 5.04e3T^{2} \)
73 \( 1 + 53.1iT - 5.32e3T^{2} \)
79 \( 1 + 23.3T + 6.24e3T^{2} \)
83 \( 1 - 102.T + 6.88e3T^{2} \)
89 \( 1 - 88.5iT - 7.92e3T^{2} \)
97 \( 1 - 140. iT - 9.40e3T^{2} \)
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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

−10.71878486724361350316797819617, −9.647655605817556295156387187131, −8.657882137836429564795279164561, −8.292652366028348145478989867887, −7.40839754623047032768004232005, −5.62907700053456702729161964841, −3.97526038467930011566371399268, −3.42434673606227458224047467168, −2.29790147063534417534879900391, −0.40659777709482573405162545788, 1.94356811341896219499354614889, 3.76327115467846482290184821823, 4.48926729180824389295683797544, 6.01065133543007991277293235992, 7.17050171364935812362664702348, 8.007431927665736222867431766435, 8.372545651630064870670228901301, 9.584311742379709779006732314454, 10.11612585054918032135186311579, 11.55182714826223065966635294402

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