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Properties

Label 2-392-392.261-c1-0-6
Degree $2$
Conductor $392$
Sign $-0.665 - 0.746i$
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  + (−1.19 + 0.755i)2-s + (0.912 − 0.0683i)3-s + (0.859 − 1.80i)4-s + (1.70 + 2.50i)5-s + (−1.03 + 0.770i)6-s + (−2.63 − 0.210i)7-s + (0.334 + 2.80i)8-s + (−2.13 + 0.322i)9-s + (−3.93 − 1.70i)10-s + (−0.926 + 6.14i)11-s + (0.660 − 1.70i)12-s + (−3.58 − 2.85i)13-s + (3.31 − 1.73i)14-s + (1.72 + 2.16i)15-s + (−2.52 − 3.10i)16-s + (−1.96 + 0.605i)17-s + ⋯
L(s)  = 1  + (−0.845 + 0.533i)2-s + (0.526 − 0.0394i)3-s + (0.429 − 0.902i)4-s + (0.764 + 1.12i)5-s + (−0.424 + 0.314i)6-s + (−0.996 − 0.0796i)7-s + (0.118 + 0.992i)8-s + (−0.713 + 0.107i)9-s + (−1.24 − 0.539i)10-s + (−0.279 + 1.85i)11-s + (0.190 − 0.492i)12-s + (−0.994 − 0.792i)13-s + (0.885 − 0.464i)14-s + (0.446 + 0.560i)15-s + (−0.630 − 0.776i)16-s + (−0.476 + 0.146i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.665 - 0.746i$
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: \((2,\ 392,\ (\ :1/2),\ -0.665 - 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.348300 + 0.777553i\)
\(L(\frac12)\) \(\approx\) \(0.348300 + 0.777553i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 - 0.755i)T \)
7 \( 1 + (2.63 + 0.210i)T \)
good3 \( 1 + (-0.912 + 0.0683i)T + (2.96 - 0.447i)T^{2} \)
5 \( 1 + (-1.70 - 2.50i)T + (-1.82 + 4.65i)T^{2} \)
11 \( 1 + (0.926 - 6.14i)T + (-10.5 - 3.24i)T^{2} \)
13 \( 1 + (3.58 + 2.85i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (1.96 - 0.605i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-6.08 - 3.51i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.93 - 2.13i)T + (19.0 + 12.9i)T^{2} \)
29 \( 1 + (5.68 + 1.29i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + (-0.241 - 0.418i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.89 + 3.12i)T + (-2.76 - 36.8i)T^{2} \)
41 \( 1 + (-3.03 - 1.45i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-2.23 - 4.64i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (1.05 + 2.69i)T + (-34.4 + 31.9i)T^{2} \)
53 \( 1 + (1.11 + 1.20i)T + (-3.96 + 52.8i)T^{2} \)
59 \( 1 + (1.10 - 1.61i)T + (-21.5 - 54.9i)T^{2} \)
61 \( 1 + (4.69 - 5.05i)T + (-4.55 - 60.8i)T^{2} \)
67 \( 1 + (-5.81 + 3.35i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.676 - 2.96i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (0.819 - 2.08i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (0.293 - 0.508i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.82 + 4.64i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (-1.80 + 0.271i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−11.30103584899627799239482376588, −10.26002156180407059527296752548, −9.749448914159512976177077781997, −9.197558393999200639638923367405, −7.51684906079482376797136287178, −7.33313617724656079183294584433, −6.15738723090237286754850562363, −5.21107244920508014122968518495, −3.03357059268439717144976542593, −2.21233201407130609975473516126, 0.65707538189631499794160369673, 2.53527077547875537731716540103, 3.36402060146253295176785104429, 5.13352429308390585667938857812, 6.27041572121694093303076910155, 7.49710527619147249324884505314, 8.798149036034219586316409560664, 9.078359305201863996332591188050, 9.627601061807911789888724411878, 10.97920272211206851131342453817

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