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Properties

Label 2-392-56.51-c2-0-45
Degree $2$
Conductor $392$
Sign $-0.351 + 0.936i$
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  + (−1.56 + 1.23i)2-s + (0.0487 + 0.0843i)3-s + (0.925 − 3.89i)4-s + (−3.00 − 1.73i)5-s + (−0.181 − 0.0720i)6-s + (3.37 + 7.25i)8-s + (4.49 − 7.78i)9-s + (6.85 − 1.00i)10-s + (1.46 + 2.53i)11-s + (0.373 − 0.111i)12-s + 19.1i·13-s − 0.337i·15-s + (−14.2 − 7.20i)16-s + (−7.19 − 12.4i)17-s + (2.59 + 17.7i)18-s + (4.04 − 7.01i)19-s + ⋯
L(s)  = 1  + (−0.784 + 0.619i)2-s + (0.0162 + 0.0281i)3-s + (0.231 − 0.972i)4-s + (−0.600 − 0.346i)5-s + (−0.0301 − 0.0120i)6-s + (0.421 + 0.906i)8-s + (0.499 − 0.865i)9-s + (0.685 − 0.100i)10-s + (0.133 + 0.230i)11-s + (0.0311 − 0.00928i)12-s + 1.47i·13-s − 0.0225i·15-s + (−0.892 − 0.450i)16-s + (−0.423 − 0.733i)17-s + (0.144 + 0.988i)18-s + (0.213 − 0.369i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.351 + 0.936i$
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: \((2,\ 392,\ (\ :1),\ -0.351 + 0.936i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.251655 - 0.363102i\)
\(L(\frac12)\) \(\approx\) \(0.251655 - 0.363102i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.56 - 1.23i)T \)
7 \( 1 \)
good3 \( 1 + (-0.0487 - 0.0843i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (3.00 + 1.73i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-1.46 - 2.53i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 19.1iT - 169T^{2} \)
17 \( 1 + (7.19 + 12.4i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-4.04 + 7.01i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (14.5 + 8.37i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 27.1iT - 841T^{2} \)
31 \( 1 + (38.8 - 22.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (34.2 + 19.7i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 45.8T + 1.68e3T^{2} \)
43 \( 1 - 61.0T + 1.84e3T^{2} \)
47 \( 1 + (40.0 + 23.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-8.39 + 4.84i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (57.2 + 99.2i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (6.47 + 3.74i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-6.02 - 10.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 129. iT - 5.04e3T^{2} \)
73 \( 1 + (9.14 + 15.8i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-36.9 - 21.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 109.T + 6.88e3T^{2} \)
89 \( 1 + (40.4 - 70.0i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 162.T + 9.40e3T^{2} \)
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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

−10.69099422167429827316556936202, −9.463252616600074226269924932425, −9.148044524073188486409398421494, −8.033886597884121350191152024227, −7.01494036832740248396694218054, −6.43334518271897079354671527142, −4.94650735024982654458162108041, −3.96181143297807098335770693439, −1.88904864333896710363744293001, −0.25094203447849710628670608485, 1.61445960986782808735707971370, 3.09021871289601351109091047405, 4.05909827163876604003578810809, 5.59623864255750826293832097360, 7.12146230071727558732028306120, 7.80707120136377839224842610801, 8.507979931027964777969761681268, 9.731039698071800252088176049685, 10.62414513695961814994561523125, 11.02813992435013975837926639759

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