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Properties

Label 2-392-7.4-c3-0-4
Degree $2$
Conductor $392$
Sign $-0.900 - 0.435i$
Analytic cond. $23.1287$
Root an. cond. $4.80923$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 1.22i)3-s + (7.07 + 12.2i)5-s + (12.5 + 21.6i)9-s + (−27 + 46.7i)11-s − 22.6·13-s − 20·15-s + (16.2 − 28.1i)17-s + (−31.8 − 55.1i)19-s + (−14 − 24.2i)23-s + (−37.5 + 64.9i)25-s − 73.5·27-s + 282·29-s + (−137. + 237. i)31-s + (−38.1 − 66.1i)33-s + (−73 − 126. i)37-s + ⋯
L(s)  = 1  + (−0.136 + 0.235i)3-s + (0.632 + 1.09i)5-s + (0.462 + 0.801i)9-s + (−0.740 + 1.28i)11-s − 0.482·13-s − 0.344·15-s + (0.232 − 0.401i)17-s + (−0.384 − 0.665i)19-s + (−0.126 − 0.219i)23-s + (−0.299 + 0.519i)25-s − 0.524·27-s + 1.80·29-s + (−0.794 + 1.37i)31-s + (−0.201 − 0.348i)33-s + (−0.324 − 0.561i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.900 - 0.435i$
Analytic conductor: \(23.1287\)
Root analytic conductor: \(4.80923\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :3/2),\ -0.900 - 0.435i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.377715211\)
\(L(\frac12)\) \(\approx\) \(1.377715211\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.707 - 1.22i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-7.07 - 12.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (27 - 46.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 22.6T + 2.19e3T^{2} \)
17 \( 1 + (-16.2 + 28.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (31.8 + 55.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (14 + 24.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 282T + 2.43e4T^{2} \)
31 \( 1 + (137. - 237. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (73 + 126. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 340.T + 6.89e4T^{2} \)
43 \( 1 - 10T + 7.95e4T^{2} \)
47 \( 1 + (253. + 438. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (299 - 517. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-287. + 498. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-233. - 404. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (458 - 793. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 420T + 3.57e5T^{2} \)
73 \( 1 + (351. - 608. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-146 - 252. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.15e3T + 5.71e5T^{2} \)
89 \( 1 + (-222. - 385. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 589.T + 9.12e5T^{2} \)
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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

−10.98543675015517749264255919026, −10.15567214991413348178746398110, −9.989818882844448764280663867538, −8.499167429143928724080851532496, −7.21195472363501496365356043804, −6.83427731960468538839686244688, −5.33188918448513638531202882036, −4.57507110472911769920716216049, −2.87297269028091298628261662277, −1.98399667399489176010607206870, 0.45494185190542747408264827284, 1.65661759166162115147935222532, 3.31349222672237925948131457177, 4.69167521236806134196107212270, 5.70436303209062120110441229818, 6.45200117779743976212433073586, 7.896188438994735997742915913249, 8.616945143452722096577615764300, 9.597809580419928119459188574815, 10.33220339625712162435236993919

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