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Properties

Label 2-392-56.5-c2-0-26
Degree $2$
Conductor $392$
Sign $0.136 - 0.990i$
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  + (−1.61 + 1.18i)2-s + (0.455 + 0.788i)3-s + (1.21 − 3.81i)4-s + (−3.17 + 5.49i)5-s + (−1.66 − 0.735i)6-s + (2.54 + 7.58i)8-s + (4.08 − 7.07i)9-s + (−1.36 − 12.6i)10-s + (11.4 − 6.60i)11-s + (3.55 − 0.780i)12-s + 19.4·13-s − 5.77·15-s + (−13.0 − 9.23i)16-s + (−13.7 + 7.96i)17-s + (1.76 + 16.2i)18-s + (8.22 − 14.2i)19-s + ⋯
L(s)  = 1  + (−0.807 + 0.590i)2-s + (0.151 + 0.262i)3-s + (0.302 − 0.953i)4-s + (−0.634 + 1.09i)5-s + (−0.277 − 0.122i)6-s + (0.318 + 0.948i)8-s + (0.453 − 0.786i)9-s + (−0.136 − 1.26i)10-s + (1.04 − 0.600i)11-s + (0.296 − 0.0650i)12-s + 1.49·13-s − 0.385·15-s + (−0.816 − 0.577i)16-s + (−0.811 + 0.468i)17-s + (0.0977 + 0.902i)18-s + (0.433 − 0.750i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.136 - 0.990i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.136 - 0.990i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.906858 + 0.790853i\)
\(L(\frac12)\) \(\approx\) \(0.906858 + 0.790853i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.61 - 1.18i)T \)
7 \( 1 \)
good3 \( 1 + (-0.455 - 0.788i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (3.17 - 5.49i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (-11.4 + 6.60i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 19.4T + 169T^{2} \)
17 \( 1 + (13.7 - 7.96i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-8.22 + 14.2i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (11.9 - 20.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 16.6iT - 841T^{2} \)
31 \( 1 + (-11.1 + 6.42i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-41.1 - 23.7i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 6.49iT - 1.68e3T^{2} \)
43 \( 1 - 33.2iT - 1.84e3T^{2} \)
47 \( 1 + (-18.9 - 10.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (32.2 - 18.5i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-27.3 - 47.3i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-5.12 + 8.87i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (14.8 - 8.56i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 32.0T + 5.04e3T^{2} \)
73 \( 1 + (92.8 - 53.5i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-29.1 + 50.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 36.3T + 6.88e3T^{2} \)
89 \( 1 + (0.929 + 0.536i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 169. iT - 9.40e3T^{2} \)
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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

−11.29340978596425563920381843265, −10.32959736109194144706548419754, −9.283933989975493082683914961457, −8.645073721819464196794030275244, −7.54014460950376802057040077547, −6.56931782619038632348489698921, −6.11164189579774779288245402596, −4.20963944319674493667498166256, −3.19984253623598338571961038893, −1.17047059021069097020873482148, 0.880447081404240070904149591196, 2.00705464639102490011323047553, 3.83931112591923107550689551751, 4.56161486252284203096401337787, 6.37282795981071077482567802815, 7.49220931641106808431787350603, 8.339173438581897215221228606611, 8.898950901490782470873182548858, 9.891942395290899821047558136377, 10.93867422452305307972513010079

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