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Properties

Label 2-392-392.261-c1-0-31
Degree $2$
Conductor $392$
Sign $0.623 - 0.781i$
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  + (1.16 + 0.794i)2-s + (1.44 − 0.108i)3-s + (0.736 + 1.85i)4-s + (−0.623 − 0.914i)5-s + (1.77 + 1.02i)6-s + (2.16 + 1.51i)7-s + (−0.616 + 2.76i)8-s + (−0.884 + 0.133i)9-s + (−0.00247 − 1.56i)10-s + (0.746 − 4.95i)11-s + (1.26 + 2.61i)12-s + (2.84 + 2.26i)13-s + (1.32 + 3.49i)14-s + (−1.00 − 1.25i)15-s + (−2.91 + 2.73i)16-s + (−4.03 + 1.24i)17-s + ⋯
L(s)  = 1  + (0.827 + 0.562i)2-s + (0.835 − 0.0626i)3-s + (0.368 + 0.929i)4-s + (−0.278 − 0.408i)5-s + (0.726 + 0.417i)6-s + (0.818 + 0.574i)7-s + (−0.217 + 0.975i)8-s + (−0.294 + 0.0444i)9-s + (−0.000781 − 0.494i)10-s + (0.225 − 1.49i)11-s + (0.365 + 0.753i)12-s + (0.788 + 0.628i)13-s + (0.354 + 0.935i)14-s + (−0.258 − 0.324i)15-s + (−0.728 + 0.684i)16-s + (−0.978 + 0.301i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.623 - 0.781i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ 0.623 - 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42745 + 1.16904i\)
\(L(\frac12)\) \(\approx\) \(2.42745 + 1.16904i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 - 0.794i)T \)
7 \( 1 + (-2.16 - 1.51i)T \)
good3 \( 1 + (-1.44 + 0.108i)T + (2.96 - 0.447i)T^{2} \)
5 \( 1 + (0.623 + 0.914i)T + (-1.82 + 4.65i)T^{2} \)
11 \( 1 + (-0.746 + 4.95i)T + (-10.5 - 3.24i)T^{2} \)
13 \( 1 + (-2.84 - 2.26i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (4.03 - 1.24i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-0.400 - 0.231i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.33 + 1.33i)T + (19.0 + 12.9i)T^{2} \)
29 \( 1 + (-4.34 - 0.992i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + (-0.283 - 0.491i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.64 + 6.08i)T + (-2.76 - 36.8i)T^{2} \)
41 \( 1 + (11.0 + 5.29i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (0.917 + 1.90i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-3.02 - 7.69i)T + (-34.4 + 31.9i)T^{2} \)
53 \( 1 + (8.22 + 8.86i)T + (-3.96 + 52.8i)T^{2} \)
59 \( 1 + (-1.21 + 1.78i)T + (-21.5 - 54.9i)T^{2} \)
61 \( 1 + (2.74 - 2.96i)T + (-4.55 - 60.8i)T^{2} \)
67 \( 1 + (-3.55 + 2.04i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.03 - 8.89i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-4.47 + 11.3i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (4.68 - 8.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.63 - 5.29i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (-8.39 + 1.26i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 - 6.83T + 97T^{2} \)
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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

−11.52046261339267688781010062601, −10.97788270454747752609963478534, −8.954122635510630144365313464769, −8.484777785009626858597635305620, −8.040378204598997520016031458058, −6.51015991133440003945331133922, −5.69010457988768371573091633033, −4.46595135864151076164429262959, −3.46566740711356548284700005430, −2.20022231891157078700710491063, 1.74816520255478365523214751420, 2.99152586213244179119206792744, 4.06511336916420823635297258584, 4.93988803254494198981869719741, 6.39656142608277802936541578586, 7.40907175429535089915381511860, 8.377176205345513130412842020409, 9.573193285117423289039835468061, 10.39746847105836118203069487608, 11.32801429959190434705320909523

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