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Properties

Label 2-392-7.4-c5-0-6
Degree $2$
Conductor $392$
Sign $-0.605 - 0.795i$
Analytic cond. $62.8704$
Root an. cond. $7.92908$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.152 + 0.263i)3-s + (−17.7 − 30.7i)5-s + (121. + 210. i)9-s + (−282. + 490. i)11-s + 983.·13-s + 10.8·15-s + (−100. + 173. i)17-s + (−414. − 717. i)19-s + (−2.21e3 − 3.84e3i)23-s + (931. − 1.61e3i)25-s − 147.·27-s − 3.71e3·29-s + (−496. + 859. i)31-s + (−86.0 − 149. i)33-s + (4.17e3 + 7.23e3i)37-s + ⋯
L(s)  = 1  + (−0.00975 + 0.0168i)3-s + (−0.317 − 0.550i)5-s + (0.499 + 0.865i)9-s + (−0.704 + 1.22i)11-s + 1.61·13-s + 0.0123·15-s + (−0.0839 + 0.145i)17-s + (−0.263 − 0.456i)19-s + (−0.874 − 1.51i)23-s + (0.298 − 0.516i)25-s − 0.0390·27-s − 0.820·29-s + (−0.0927 + 0.160i)31-s + (−0.0137 − 0.0238i)33-s + (0.501 + 0.869i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(62.8704\)
Root analytic conductor: \(7.92908\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :5/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9632579404\)
\(L(\frac12)\) \(\approx\) \(0.9632579404\)
\(L(\frac{7}{2})\) 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 \( 1 + (0.152 - 0.263i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (17.7 + 30.7i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (282. - 490. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 983.T + 3.71e5T^{2} \)
17 \( 1 + (100. - 173. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (414. + 717. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (2.21e3 + 3.84e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 3.71e3T + 2.05e7T^{2} \)
31 \( 1 + (496. - 859. i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-4.17e3 - 7.23e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.34e4T + 1.15e8T^{2} \)
43 \( 1 - 298.T + 1.47e8T^{2} \)
47 \( 1 + (-9.36e3 - 1.62e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (8.01e3 - 1.38e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (6.37e3 - 1.10e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.74e4 - 3.02e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (5.98e3 - 1.03e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 1.29e4T + 1.80e9T^{2} \)
73 \( 1 + (4.05e4 - 7.03e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (2.34e4 + 4.07e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.11e5T + 3.93e9T^{2} \)
89 \( 1 + (-1.73e4 - 3.00e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 9.26e4T + 8.58e9T^{2} \)
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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

−10.64479912457299850754532649747, −10.12018885845450530681899514675, −8.777286673732508467525744691787, −8.141621067913914637286013628245, −7.16713834189460143748624129601, −6.03347864139348046109979606243, −4.75453287673315543991578367108, −4.16945796471070832342631485108, −2.48650285862494597085611534268, −1.31546853763112885483514635584, 0.24499490105835093343614847017, 1.58528541862157101033574369158, 3.37285792841901014882664435243, 3.76611729739136543647823755948, 5.56157608504446877049944442993, 6.27200422067160980007821737103, 7.37383921156472908246912794409, 8.307212191726908302964814862190, 9.200377333353941144476590467734, 10.29333922633703244510159768508

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