Properties

Label 2-392-392.37-c1-0-31
Degree $2$
Conductor $392$
Sign $0.962 - 0.270i$
Analytic cond. $3.13013$
Root an. cond. $1.76921$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.23 + 0.695i)2-s + (1.73 + 1.86i)3-s + (1.03 − 1.71i)4-s + (0.880 − 2.85i)5-s + (−3.43 − 1.09i)6-s + (−1.33 − 2.28i)7-s + (−0.0799 + 2.82i)8-s + (−0.261 + 3.48i)9-s + (0.901 + 4.12i)10-s + (1.69 − 0.127i)11-s + (4.99 − 1.04i)12-s + (0.482 − 1.00i)13-s + (3.23 + 1.87i)14-s + (6.86 − 3.30i)15-s + (−1.86 − 3.53i)16-s + (2.79 + 0.420i)17-s + ⋯
L(s)  = 1  + (−0.870 + 0.491i)2-s + (1.00 + 1.07i)3-s + (0.516 − 0.856i)4-s + (0.393 − 1.27i)5-s + (−1.40 − 0.447i)6-s + (−0.505 − 0.862i)7-s + (−0.0282 + 0.999i)8-s + (−0.0871 + 1.16i)9-s + (0.284 + 1.30i)10-s + (0.511 − 0.0383i)11-s + (1.44 − 0.300i)12-s + (0.133 − 0.278i)13-s + (0.864 + 0.502i)14-s + (1.77 − 0.853i)15-s + (−0.467 − 0.884i)16-s + (0.676 + 0.102i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.962 - 0.270i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ 0.962 - 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32445 + 0.182553i\)
\(L(\frac12)\) \(\approx\) \(1.32445 + 0.182553i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 - 0.695i)T \)
7 \( 1 + (1.33 + 2.28i)T \)
good3 \( 1 + (-1.73 - 1.86i)T + (-0.224 + 2.99i)T^{2} \)
5 \( 1 + (-0.880 + 2.85i)T + (-4.13 - 2.81i)T^{2} \)
11 \( 1 + (-1.69 + 0.127i)T + (10.8 - 1.63i)T^{2} \)
13 \( 1 + (-0.482 + 1.00i)T + (-8.10 - 10.1i)T^{2} \)
17 \( 1 + (-2.79 - 0.420i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (-0.392 - 0.226i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.01 + 1.05i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (6.88 + 5.48i)T + (6.45 + 28.2i)T^{2} \)
31 \( 1 + (-4.94 - 8.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.08 - 3.56i)T + (27.1 - 25.1i)T^{2} \)
41 \( 1 + (0.419 + 1.83i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-3.27 - 0.748i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (2.83 - 1.93i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-7.46 - 2.92i)T + (38.8 + 36.0i)T^{2} \)
59 \( 1 + (3.14 + 10.2i)T + (-48.7 + 33.2i)T^{2} \)
61 \( 1 + (6.55 - 2.57i)T + (44.7 - 41.4i)T^{2} \)
67 \( 1 + (7.66 - 4.42i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.596 + 0.747i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (7.21 + 4.92i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (4.98 - 8.63i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.02 - 2.12i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (1.13 - 15.1i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + 9.10T + 97T^{2} \)
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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

−10.78150097592206111487547326438, −10.06951933520007431787620848965, −9.337382602943838541114127947735, −8.839194551030266724254671461315, −8.006454660729332798809302914166, −6.83508420267973783414606306745, −5.45898725880828988384373873928, −4.48470101450524594083075116229, −3.18794041434077403549397224200, −1.22608780737369617512665911719, 1.71411451754273101065978365403, 2.74646832480406017938705423915, 3.38589084355272988365109696624, 6.01815756985340420419874873123, 7.00505344542925428527911818674, 7.43039037676441052176954949959, 8.721908016943280566846337990596, 9.234063805244445894131363157600, 10.21915528104903507756528903549, 11.26238081207137569528050823051

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