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Properties

Label 2-392-56.51-c2-0-31
Degree $2$
Conductor $392$
Sign $0.526 - 0.850i$
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  + (1.99 − 0.0637i)2-s + (1.72 + 2.98i)3-s + (3.99 − 0.254i)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 + (−8.60 − 4.61i)10-s + (10.7 + 18.6i)11-s + (7.62 + 11.4i)12-s + 13.0i·13-s − 16.8i·15-s + (15.8 − 2.03i)16-s + (0.117 + 0.203i)17-s + (−2.68 + 5.01i)18-s + (−2.27 + 3.94i)19-s + ⋯
L(s)  = 1  + (0.999 − 0.0318i)2-s + (0.573 + 0.993i)3-s + (0.997 − 0.0636i)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 + (−0.860 − 0.461i)10-s + (0.976 + 1.69i)11-s + (0.635 + 0.954i)12-s + 1.00i·13-s − 1.12i·15-s + (0.991 − 0.127i)16-s + (0.00690 + 0.0119i)17-s + (−0.149 + 0.278i)18-s + (−0.119 + 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.526 - 0.850i$
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: \((2,\ 392,\ (\ :1),\ 0.526 - 0.850i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.12862 + 1.74190i\)
\(L(\frac12)\) \(\approx\) \(3.12862 + 1.74190i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 + 0.0637i)T \)
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

−11.70435671476906006948042525517, −10.23017781947977438737777985339, −9.578605923212410470670073750468, −8.534670009284048581362578902764, −7.34755179902519378321035482618, −6.48172028969783132868963429917, −4.77283962132748850602342159906, −4.31295932427434663931390648594, −3.57097447539988547782349944703, −1.91441563501344897825974465074, 1.24832905047075442633906207119, 3.01091928139247381984370854324, 3.48451031624468075834170544758, 5.06910932917739261078217554360, 6.38932921956408026988514165539, 6.99565852689369600007825098178, 8.026912978224385486433390308939, 8.601576204338132482533799037241, 10.46156318981350943284585591583, 11.23740967142271460336258544367

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