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Properties

Label 2-392-7.2-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  + (−7.78 − 13.4i)3-s + (−48.3 + 83.7i)5-s + (0.222 − 0.385i)9-s + (−140. − 243. i)11-s − 269.·13-s + 1.50e3·15-s + (−859. − 1.48e3i)17-s + (586. − 1.01e3i)19-s + (−392. + 680. i)23-s + (−3.11e3 − 5.39e3i)25-s − 3.79e3·27-s + 6.14e3·29-s + (3.50e3 + 6.06e3i)31-s + (−2.19e3 + 3.79e3i)33-s + (5.74e3 − 9.95e3i)37-s + ⋯
L(s)  = 1  + (−0.499 − 0.865i)3-s + (−0.865 + 1.49i)5-s + (0.000915 − 0.00158i)9-s + (−0.350 − 0.607i)11-s − 0.442·13-s + 1.72·15-s + (−0.721 − 1.24i)17-s + (0.372 − 0.645i)19-s + (−0.154 + 0.268i)23-s + (−0.996 − 1.72i)25-s − 1.00·27-s + 1.35·29-s + (0.654 + 1.13i)31-s + (−0.350 + 0.606i)33-s + (0.690 − 1.19i)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: $\chi_{392} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :5/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7072680530\)
\(L(\frac12)\) \(\approx\) \(0.7072680530\)
\(L(\frac{7}{2})\) 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 \( 1 + (7.78 + 13.4i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (48.3 - 83.7i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (140. + 243. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 269.T + 3.71e5T^{2} \)
17 \( 1 + (859. + 1.48e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-586. + 1.01e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (392. - 680. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 6.14e3T + 2.05e7T^{2} \)
31 \( 1 + (-3.50e3 - 6.06e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-5.74e3 + 9.95e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 1.32e4T + 1.15e8T^{2} \)
43 \( 1 + 1.98e4T + 1.47e8T^{2} \)
47 \( 1 + (3.44e3 - 5.96e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (4.32e3 + 7.49e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-2.39e4 - 4.14e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.63e4 - 4.56e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.20e4 - 2.08e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 1.25e4T + 1.80e9T^{2} \)
73 \( 1 + (2.03e3 + 3.53e3i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-5.95e3 + 1.03e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 8.19e4T + 3.93e9T^{2} \)
89 \( 1 + (-4.84e4 + 8.39e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 2.64e4T + 8.58e9T^{2} \)
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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

−10.83892630307619312629314525923, −9.947963225957462328386758863429, −8.573270198553043582696082663125, −7.40740787699161845879858920679, −6.99586465052497860550002584242, −6.21766328426706808064398372498, −4.81192624683705689586409347547, −3.34907069927381515925146530819, −2.50940332201323443830629544799, −0.71976169260979511767510170746, 0.28439532405569731709139719854, 1.76424751148233934640228884646, 3.72255891347403556700597125848, 4.69356383387818377479162549106, 4.99164002277475659554360977239, 6.40514625244846547297769601392, 7.962244713326532970358988943240, 8.346396866502072231491340485274, 9.643619208207688000977236878517, 10.20699699497143824317197514874

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