Properties

Label 2-392-56.11-c2-0-9
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: $\chi_{392} (67, \cdot )$
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

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

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