Properties

Label 2-392-392.109-c1-0-36
Degree $2$
Conductor $392$
Sign $-0.797 + 0.602i$
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  + (−0.569 − 1.29i)2-s + (−0.443 − 2.93i)3-s + (−1.35 + 1.47i)4-s + (3.67 + 1.44i)5-s + (−3.55 + 2.24i)6-s + (2.36 − 1.18i)7-s + (2.67 + 0.911i)8-s + (−5.57 + 1.72i)9-s + (−0.224 − 5.58i)10-s + (1.30 − 4.24i)11-s + (4.93 + 3.32i)12-s + (1.69 − 0.387i)13-s + (−2.88 − 2.38i)14-s + (2.61 − 11.4i)15-s + (−0.344 − 3.98i)16-s + (0.673 − 0.459i)17-s + ⋯
L(s)  = 1  + (−0.402 − 0.915i)2-s + (−0.255 − 1.69i)3-s + (−0.675 + 0.736i)4-s + (1.64 + 0.645i)5-s + (−1.45 + 0.917i)6-s + (0.893 − 0.449i)7-s + (0.946 + 0.322i)8-s + (−1.85 + 0.573i)9-s + (−0.0710 − 1.76i)10-s + (0.394 − 1.27i)11-s + (1.42 + 0.958i)12-s + (0.471 − 0.107i)13-s + (−0.770 − 0.636i)14-s + (0.674 − 2.95i)15-s + (−0.0861 − 0.996i)16-s + (0.163 − 0.111i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 + 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.797 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.797 + 0.602i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ -0.797 + 0.602i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.431669 - 1.28767i\)
\(L(\frac12)\) \(\approx\) \(0.431669 - 1.28767i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.569 + 1.29i)T \)
7 \( 1 + (-2.36 + 1.18i)T \)
good3 \( 1 + (0.443 + 2.93i)T + (-2.86 + 0.884i)T^{2} \)
5 \( 1 + (-3.67 - 1.44i)T + (3.66 + 3.40i)T^{2} \)
11 \( 1 + (-1.30 + 4.24i)T + (-9.08 - 6.19i)T^{2} \)
13 \( 1 + (-1.69 + 0.387i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (-0.673 + 0.459i)T + (6.21 - 15.8i)T^{2} \)
19 \( 1 + (5.68 - 3.27i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.85 - 2.63i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (2.32 - 4.82i)T + (-18.0 - 22.6i)T^{2} \)
31 \( 1 + (0.0831 - 0.144i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.25 - 0.468i)T + (36.5 - 5.51i)T^{2} \)
41 \( 1 + (1.52 + 1.91i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (1.43 + 1.14i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (5.80 - 5.38i)T + (3.51 - 46.8i)T^{2} \)
53 \( 1 + (1.35 + 0.101i)T + (52.4 + 7.89i)T^{2} \)
59 \( 1 + (5.50 - 2.16i)T + (43.2 - 40.1i)T^{2} \)
61 \( 1 + (-7.34 + 0.550i)T + (60.3 - 9.09i)T^{2} \)
67 \( 1 + (-1.43 - 0.830i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.00 - 0.966i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (2.69 + 2.50i)T + (5.45 + 72.7i)T^{2} \)
79 \( 1 + (1.37 + 2.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.52 + 1.26i)T + (74.7 + 36.0i)T^{2} \)
89 \( 1 + (3.20 - 0.988i)T + (73.5 - 50.1i)T^{2} \)
97 \( 1 - 16.3T + 97T^{2} \)
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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

−10.96693522696503188769925742585, −10.41386416808878653693781515554, −8.985371612691793867267515946946, −8.290697455906135060700711124628, −7.19145488035334642060083241397, −6.27492276589669738580000269214, −5.36701622109731326213069577112, −3.25382447444428191578296013691, −1.94139417235422540147953606727, −1.27667605642692489412633480052, 1.91901114654736777773757503982, 4.41129259745353506444596725545, 4.91556308223093204498427851939, 5.70312796101769555331110095955, 6.66156535180761345902101030519, 8.502514447003302530440904384174, 8.999569501928437356616395370883, 9.729665124407780868423253948878, 10.32500160843388242477204783073, 11.20846627927925069570871722896

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