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Properties

Label 2-392-7.2-c5-0-2
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  + (−10 − 17.3i)3-s + (37 − 64.0i)5-s + (−78.5 + 135. i)9-s + (−62 − 107. i)11-s + 478·13-s − 1.48e3·15-s + (599 + 1.03e3i)17-s + (−1.52e3 + 2.63e3i)19-s + (−92 + 159. i)23-s + (−1.17e3 − 2.03e3i)25-s − 1.71e3·27-s − 3.28e3·29-s + (2.86e3 + 4.96e3i)31-s + (−1.24e3 + 2.14e3i)33-s + (−5.16e3 + 8.94e3i)37-s + ⋯
L(s)  = 1  + (−0.641 − 1.11i)3-s + (0.661 − 1.14i)5-s + (−0.323 + 0.559i)9-s + (−0.154 − 0.267i)11-s + 0.784·13-s − 1.69·15-s + (0.502 + 0.870i)17-s + (−0.967 + 1.67i)19-s + (−0.0362 + 0.0628i)23-s + (−0.376 − 0.651i)25-s − 0.454·27-s − 0.724·29-s + (0.535 + 0.927i)31-s + (−0.198 + 0.343i)33-s + (−0.620 + 1.07i)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

\(L(3)\) \(\approx\) \(0.6363756309\)
\(L(\frac12)\) \(\approx\) \(0.6363756309\)
\(L(\frac{7}{2})\) 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 \( 1 + (10 + 17.3i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-37 + 64.0i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (62 + 107. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 478T + 3.71e5T^{2} \)
17 \( 1 + (-599 - 1.03e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.52e3 - 2.63e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (92 - 159. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 3.28e3T + 2.05e7T^{2} \)
31 \( 1 + (-2.86e3 - 4.96e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (5.16e3 - 8.94e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 8.88e3T + 1.15e8T^{2} \)
43 \( 1 + 9.18e3T + 1.47e8T^{2} \)
47 \( 1 + (1.18e4 - 2.04e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (5.84e3 + 1.01e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (8.43e3 + 1.46e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-9.24e3 + 1.60e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-7.76e3 - 1.34e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 3.19e4T + 1.80e9T^{2} \)
73 \( 1 + (-2.44e3 - 4.23e3i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (2.22e4 - 3.85e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 6.73e4T + 3.93e9T^{2} \)
89 \( 1 + (3.59e4 - 6.23e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 4.88e4T + 8.58e9T^{2} \)
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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

−10.66783080981340879450835246345, −9.746723394328253757376668773537, −8.494448480279931156470561034938, −8.031998525188269523348244656781, −6.54713160563795835363331232449, −5.97738547920479228135702661687, −5.10952667923917946020897897230, −3.65707688917403336997711861529, −1.65532519806933513359798353841, −1.30409758816197589469217384022, 0.16995465515635073577894450080, 2.17455745708859994782932089886, 3.33674395205035168783289064063, 4.52794224933280397978147364901, 5.49035294308262908639791900610, 6.42987583563835415425886580191, 7.30518112164259140487893381265, 8.814920390660059116567004800771, 9.742208080224699266486639568271, 10.40915354379092878042248877332

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