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Properties

Label 2-392-7.4-c5-0-42
Degree $2$
Conductor $392$
Sign $-0.701 + 0.712i$
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  + (3.44 − 5.96i)3-s + (−24.2 − 41.9i)5-s + (97.7 + 169. i)9-s + (81.7 − 141. i)11-s + 120.·13-s − 334.·15-s + (39.0 − 67.7i)17-s + (−1.13e3 − 1.96e3i)19-s + (1.22e3 + 2.12e3i)23-s + (388. − 672. i)25-s + 3.02e3·27-s + 6.98e3·29-s + (1.39e3 − 2.41e3i)31-s + (−563. − 975. i)33-s + (−4.72e3 − 8.19e3i)37-s + ⋯
L(s)  = 1  + (0.221 − 0.382i)3-s + (−0.433 − 0.750i)5-s + (0.402 + 0.696i)9-s + (0.203 − 0.352i)11-s + 0.198·13-s − 0.383·15-s + (0.0328 − 0.0568i)17-s + (−0.719 − 1.24i)19-s + (0.483 + 0.836i)23-s + (0.124 − 0.215i)25-s + 0.797·27-s + 1.54·29-s + (0.261 − 0.452i)31-s + (−0.0900 − 0.156i)33-s + (−0.567 − 0.983i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.701 + 0.712i$
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: \((2,\ 392,\ (\ :5/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.563869391\)
\(L(\frac12)\) \(\approx\) \(1.563869391\)
\(L(\frac{7}{2})\) 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 \( 1 + (-3.44 + 5.96i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (24.2 + 41.9i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-81.7 + 141. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 120.T + 3.71e5T^{2} \)
17 \( 1 + (-39.0 + 67.7i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (1.13e3 + 1.96e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.22e3 - 2.12e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 6.98e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.39e3 + 2.41e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (4.72e3 + 8.19e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.00e4T + 1.15e8T^{2} \)
43 \( 1 + 6.93e3T + 1.47e8T^{2} \)
47 \( 1 + (-582. - 1.00e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (4.28e3 - 7.41e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-3.11e3 + 5.38e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (2.09e4 + 3.63e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (906. - 1.56e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 5.68e4T + 1.80e9T^{2} \)
73 \( 1 + (2.21e4 - 3.83e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (1.74e4 + 3.02e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 3.96e4T + 3.93e9T^{2} \)
89 \( 1 + (6.31e4 + 1.09e5i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.45e5T + 8.58e9T^{2} \)
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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

−10.19520070824472397095392956892, −8.942965788973123336702822939171, −8.390100100970280933147336952932, −7.41646816229496811039808207563, −6.47954385666707943339723496771, −5.09157311404654653558856690389, −4.32959971510913300062754547327, −2.89783672499759581260733792532, −1.55946663897306316582593122177, −0.39612042339703032476153738517, 1.30918355503709733049173593979, 2.91003276962842405410473691899, 3.79505440413153516082231864169, 4.77968989218570865173804028144, 6.36025758956918400473141341184, 6.91658006627477802402182033268, 8.157193927626928290606992990671, 8.962108214670993252866717958063, 10.21029388536058687921306518056, 10.48934112427184071018720906316

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