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Properties

Label 2-392-8.3-c2-0-54
Degree $2$
Conductor $392$
Sign $-0.156 + 0.987i$
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  + (1.67 + 1.08i)2-s − 3.98·3-s + (1.62 + 3.65i)4-s − 1.88i·5-s + (−6.67 − 4.33i)6-s + (−1.25 + 7.90i)8-s + 6.84·9-s + (2.05 − 3.15i)10-s − 7.87·11-s + (−6.47 − 14.5i)12-s − 11.4i·13-s + 7.49i·15-s + (−10.7 + 11.8i)16-s − 2.89·17-s + (11.4 + 7.46i)18-s − 30.0·19-s + ⋯
L(s)  = 1  + (0.838 + 0.544i)2-s − 1.32·3-s + (0.406 + 0.913i)4-s − 0.376i·5-s + (−1.11 − 0.722i)6-s + (−0.156 + 0.987i)8-s + 0.760·9-s + (0.205 − 0.315i)10-s − 0.716·11-s + (−0.539 − 1.21i)12-s − 0.883i·13-s + 0.499i·15-s + (−0.669 + 0.742i)16-s − 0.170·17-s + (0.638 + 0.414i)18-s − 1.58·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ -0.156 + 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.334818 - 0.392096i\)
\(L(\frac12)\) \(\approx\) \(0.334818 - 0.392096i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.67 - 1.08i)T \)
7 \( 1 \)
good3 \( 1 + 3.98T + 9T^{2} \)
5 \( 1 + 1.88iT - 25T^{2} \)
11 \( 1 + 7.87T + 121T^{2} \)
13 \( 1 + 11.4iT - 169T^{2} \)
17 \( 1 + 2.89T + 289T^{2} \)
19 \( 1 + 30.0T + 361T^{2} \)
23 \( 1 + 38.5iT - 529T^{2} \)
29 \( 1 + 27.8iT - 841T^{2} \)
31 \( 1 + 22.4iT - 961T^{2} \)
37 \( 1 + 45.5iT - 1.36e3T^{2} \)
41 \( 1 + 40.6T + 1.68e3T^{2} \)
43 \( 1 + 47.2T + 1.84e3T^{2} \)
47 \( 1 - 82.5iT - 2.20e3T^{2} \)
53 \( 1 - 26.8iT - 2.80e3T^{2} \)
59 \( 1 + 10.4T + 3.48e3T^{2} \)
61 \( 1 + 22.1iT - 3.72e3T^{2} \)
67 \( 1 + 59.3T + 4.48e3T^{2} \)
71 \( 1 - 38.2iT - 5.04e3T^{2} \)
73 \( 1 - 13.9T + 5.32e3T^{2} \)
79 \( 1 - 51.1iT - 6.24e3T^{2} \)
83 \( 1 + 89.4T + 6.88e3T^{2} \)
89 \( 1 - 105.T + 7.92e3T^{2} \)
97 \( 1 - 55.3T + 9.40e3T^{2} \)
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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

−10.91381474439833004752301500166, −10.43207612491201236992765621829, −8.702296768625791326752322255728, −7.87709025916652413215502439602, −6.60928128793760048567503705714, −5.98261420426425792610906216568, −5.04082492088369032554833452244, −4.31967663035974375504380985143, −2.58886572035820390631975821054, −0.19171284642439009926403476298, 1.71455932388211848167665923543, 3.29539842760384300629506882072, 4.68182728085364477816686913588, 5.35403050255698098666354561228, 6.45991659452227805384120539603, 6.98547028675156338400677816449, 8.734903702992291549638299279850, 10.14335651818982401510857040007, 10.64716168831979160693462250268, 11.46369544911954051086391255074

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