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Properties

Label 2-392-7.4-c5-0-9
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  + (−10.4 + 18.0i)3-s + (45.2 + 78.3i)5-s + (−96.7 − 167. i)9-s + (276. − 478. i)11-s + 593.·13-s − 1.88e3·15-s + (−711. + 1.23e3i)17-s + (159. + 276. i)19-s + (−329. − 571. i)23-s + (−2.52e3 + 4.38e3i)25-s − 1.03e3·27-s − 8.18e3·29-s + (−4.79e3 + 8.31e3i)31-s + (5.77e3 + 9.99e3i)33-s + (−2.59e3 − 4.48e3i)37-s + ⋯
L(s)  = 1  + (−0.670 + 1.16i)3-s + (0.809 + 1.40i)5-s + (−0.398 − 0.689i)9-s + (0.688 − 1.19i)11-s + 0.973·13-s − 2.16·15-s + (−0.596 + 1.03i)17-s + (0.101 + 0.175i)19-s + (−0.130 − 0.225i)23-s + (−0.809 + 1.40i)25-s − 0.273·27-s − 1.80·29-s + (−0.896 + 1.55i)31-s + (0.922 + 1.59i)33-s + (−0.311 − 0.538i)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

\(L(3)\) \(\approx\) \(1.020239814\)
\(L(\frac12)\) \(\approx\) \(1.020239814\)
\(L(\frac{7}{2})\) 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 \( 1 + (10.4 - 18.0i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-45.2 - 78.3i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-276. + 478. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 593.T + 3.71e5T^{2} \)
17 \( 1 + (711. - 1.23e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-159. - 276. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (329. + 571. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 8.18e3T + 2.05e7T^{2} \)
31 \( 1 + (4.79e3 - 8.31e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (2.59e3 + 4.48e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 2.19e3T + 1.15e8T^{2} \)
43 \( 1 - 7.45e3T + 1.47e8T^{2} \)
47 \( 1 + (9.78e3 + 1.69e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.82e4 - 3.16e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-8.18e3 + 1.41e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (5.44e3 + 9.43e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (4.01e3 - 6.95e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 5.59e4T + 1.80e9T^{2} \)
73 \( 1 + (3.88e4 - 6.73e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.60e3 - 2.77e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 7.66e4T + 3.93e9T^{2} \)
89 \( 1 + (4.21e4 + 7.29e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.01e5T + 8.58e9T^{2} \)
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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

−11.02238839416297704744178160891, −10.41292583265931734044591846935, −9.448086739148611098967272332397, −8.584345277985256009749924953032, −7.00264288902370642121512319030, −6.04072480128860762078961156192, −5.61104213530910466443930509422, −3.96102334675055076165089799398, −3.32323571263471906269810874825, −1.71431290194016903400674639007, 0.26850205419717432832525254104, 1.37405052981554351822529759785, 1.98290581171352384188964477430, 4.15883378450407530532515900227, 5.26173646569606810743191856565, 6.05633647402421959724450994244, 6.97932660174635313548751963682, 7.916942976144896500170102033232, 9.269361995742216383047814932597, 9.494050472975428700818732835274

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