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Properties

Label 2-392-7.2-c5-0-21
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  + (15 + 25.9i)3-s + (16 − 27.7i)5-s + (−328.5 + 568. i)9-s + (312 + 540. i)11-s + 708·13-s + 960·15-s + (467 + 808. i)17-s + (929 − 1.60e3i)19-s + (560 − 969. i)23-s + (1.05e3 + 1.81e3i)25-s − 1.24e4·27-s − 1.17e3·29-s + (1.45e3 + 2.51e3i)31-s + (−9.36e3 + 1.62e4i)33-s + (6.23e3 − 1.07e4i)37-s + ⋯
L(s)  = 1  + (0.962 + 1.66i)3-s + (0.286 − 0.495i)5-s + (−1.35 + 2.34i)9-s + (0.777 + 1.34i)11-s + 1.16·13-s + 1.10·15-s + (0.391 + 0.678i)17-s + (0.590 − 1.02i)19-s + (0.220 − 0.382i)23-s + (0.336 + 0.582i)25-s − 3.27·27-s − 0.259·29-s + (0.271 + 0.470i)31-s + (−1.49 + 2.59i)33-s + (0.748 − 1.29i)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: $\chi_{392} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :5/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.471743728\)
\(L(\frac12)\) \(\approx\) \(3.471743728\)
\(L(\frac{7}{2})\) 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 \( 1 + (-15 - 25.9i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-16 + 27.7i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-312 - 540. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 708T + 3.71e5T^{2} \)
17 \( 1 + (-467 - 808. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-929 + 1.60e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-560 + 969. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 1.17e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.45e3 - 2.51e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-6.23e3 + 1.07e4i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 2.66e3T + 1.15e8T^{2} \)
43 \( 1 + 7.14e3T + 1.47e8T^{2} \)
47 \( 1 + (3.73e3 - 6.46e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.36e4 - 2.36e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.24e3 - 2.15e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (5.54e3 - 9.60e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.98e4 + 3.44e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 6.98e4T + 1.80e9T^{2} \)
73 \( 1 + (-8.22e3 - 1.42e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (3.91e4 - 6.78e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 1.09e5T + 3.93e9T^{2} \)
89 \( 1 + (2.84e4 - 4.93e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.15e5T + 8.58e9T^{2} \)
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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

−10.59094767246149822484017994077, −9.727390410789578591845262431251, −9.099843257146321143380638974532, −8.529984197786673124661277343918, −7.28605677803584097834178864814, −5.72489597203325049824351645442, −4.67713129464723473405988880085, −4.01136531546143074472315228788, −2.92967951215178428732109902065, −1.54309185522367095235590273354, 0.792491763590318529497716301782, 1.56768343715656887228263965510, 2.95253941380448723378189919022, 3.55327563028368203396209140894, 5.89652998145406256732984954848, 6.38705977097259824679989232719, 7.36741254285767144816888849207, 8.309740891849449803749390683242, 8.837458445065026202240993100383, 9.990966490986119600583242081318

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