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Properties

Label 2-392-56.13-c2-0-50
Degree $2$
Conductor $392$
Sign $0.997 + 0.0746i$
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  + (−1.95 + 0.422i)2-s + 3.86·3-s + (3.64 − 1.65i)4-s + 4.67·5-s + (−7.56 + 1.63i)6-s + (−6.42 + 4.77i)8-s + 5.97·9-s + (−9.14 + 1.97i)10-s − 14.5i·11-s + (14.0 − 6.39i)12-s + 12.7·13-s + 18.1·15-s + (10.5 − 12.0i)16-s + 19.5i·17-s + (−11.6 + 2.52i)18-s + 17.7·19-s + ⋯
L(s)  = 1  + (−0.977 + 0.211i)2-s + 1.28·3-s + (0.910 − 0.413i)4-s + 0.935·5-s + (−1.26 + 0.272i)6-s + (−0.802 + 0.596i)8-s + 0.663·9-s + (−0.914 + 0.197i)10-s − 1.32i·11-s + (1.17 − 0.533i)12-s + 0.977·13-s + 1.20·15-s + (0.658 − 0.752i)16-s + 1.14i·17-s + (−0.648 + 0.140i)18-s + 0.932·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0746i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.997 + 0.0746i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.997 + 0.0746i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.06333 - 0.0770705i\)
\(L(\frac12)\) \(\approx\) \(2.06333 - 0.0770705i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.95 - 0.422i)T \)
7 \( 1 \)
good3 \( 1 - 3.86T + 9T^{2} \)
5 \( 1 - 4.67T + 25T^{2} \)
11 \( 1 + 14.5iT - 121T^{2} \)
13 \( 1 - 12.7T + 169T^{2} \)
17 \( 1 - 19.5iT - 289T^{2} \)
19 \( 1 - 17.7T + 361T^{2} \)
23 \( 1 - 8.86T + 529T^{2} \)
29 \( 1 + 35.4iT - 841T^{2} \)
31 \( 1 + 29.0iT - 961T^{2} \)
37 \( 1 - 12.2iT - 1.36e3T^{2} \)
41 \( 1 - 22.0iT - 1.68e3T^{2} \)
43 \( 1 - 79.8iT - 1.84e3T^{2} \)
47 \( 1 + 42.1iT - 2.20e3T^{2} \)
53 \( 1 - 36.1iT - 2.80e3T^{2} \)
59 \( 1 + 2.40T + 3.48e3T^{2} \)
61 \( 1 - 29.2T + 3.72e3T^{2} \)
67 \( 1 - 40.6iT - 4.48e3T^{2} \)
71 \( 1 + 22.6T + 5.04e3T^{2} \)
73 \( 1 + 76.3iT - 5.32e3T^{2} \)
79 \( 1 - 136.T + 6.24e3T^{2} \)
83 \( 1 + 49.9T + 6.88e3T^{2} \)
89 \( 1 - 1.12iT - 7.92e3T^{2} \)
97 \( 1 - 158. iT - 9.40e3T^{2} \)
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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

−10.83158815608313617959447318627, −9.795900352917887906217357378454, −9.203069286172961293975571299779, −8.325272047527358500539847944188, −7.86962897818802692727590143459, −6.31467069486345138546566034374, −5.75716081995221145019694199493, −3.58434203835612060633884194075, −2.54094943417472579101495985748, −1.28976481229199723696431499122, 1.51589869810938864927689893412, 2.50284321853804923799417011763, 3.55463528115962194972078545856, 5.35992555738865483978513021033, 6.84195513191464078641063949173, 7.50478734824362496846649437297, 8.642440527593917516721074014821, 9.272974998103289897943084210399, 9.828527423439527239722394003025, 10.76468417921489225320919060040

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