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Properties

Label 2-392-56.5-c2-0-63
Degree $2$
Conductor $392$
Sign $0.454 + 0.890i$
Analytic cond. $10.6812$
Root an. cond. $3.26821$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.34 − 1.48i)2-s + (1.93 + 3.35i)3-s + (−0.389 − 3.98i)4-s + (2.33 − 4.05i)5-s + (7.56 + 1.63i)6-s + (−6.42 − 4.77i)8-s + (−2.98 + 5.17i)9-s + (−2.85 − 8.90i)10-s + (12.6 − 7.29i)11-s + (12.5 − 9.00i)12-s − 12.7·13-s + 18.1·15-s + (−15.6 + 3.10i)16-s + (16.9 − 9.76i)17-s + (3.65 + 11.3i)18-s + (8.86 − 15.3i)19-s + ⋯
L(s)  = 1  + (0.671 − 0.740i)2-s + (0.644 + 1.11i)3-s + (−0.0974 − 0.995i)4-s + (0.467 − 0.810i)5-s + (1.26 + 0.272i)6-s + (−0.802 − 0.596i)8-s + (−0.331 + 0.575i)9-s + (−0.285 − 0.890i)10-s + (1.14 − 0.663i)11-s + (1.04 − 0.750i)12-s − 0.977·13-s + 1.20·15-s + (−0.981 + 0.193i)16-s + (0.994 − 0.574i)17-s + (0.202 + 0.632i)18-s + (0.466 − 0.807i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.454 + 0.890i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.454 + 0.890i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.72343 - 1.66839i\)
\(L(\frac12)\) \(\approx\) \(2.72343 - 1.66839i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 + 1.48i)T \)
7 \( 1 \)
good3 \( 1 + (-1.93 - 3.35i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-2.33 + 4.05i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (-12.6 + 7.29i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 12.7T + 169T^{2} \)
17 \( 1 + (-16.9 + 9.76i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-8.86 + 15.3i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (4.43 - 7.67i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 35.4iT - 841T^{2} \)
31 \( 1 + (25.1 - 14.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-10.5 - 6.10i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 22.0iT - 1.68e3T^{2} \)
43 \( 1 + 79.8iT - 1.84e3T^{2} \)
47 \( 1 + (-36.5 - 21.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (31.3 - 18.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (1.20 + 2.08i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-14.6 + 25.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (35.2 - 20.3i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 22.6T + 5.04e3T^{2} \)
73 \( 1 + (66.1 - 38.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (68.4 - 118. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 49.9T + 6.88e3T^{2} \)
89 \( 1 + (0.970 + 0.560i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 158. iT - 9.40e3T^{2} \)
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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

−10.88984778543354942788483086160, −9.886487506030783809932408528577, −9.285068583086376612127584709772, −8.822401035333099823329469562532, −7.04260885985301645001681117404, −5.53155335957765842568604891996, −4.89190023460391427828242776882, −3.80855278691578470430463503157, −2.90934215143456841636628180221, −1.20332687179823547959963260610, 1.89551960126116650808740981471, 2.98485893986262970349787397176, 4.27357711123877495498508223727, 5.83679529586957418043125380981, 6.59559350565222816018658822247, 7.44185287672162564077984701636, 7.958825482820609774722519490782, 9.246403249014860281614419687967, 10.18943166464690752170909815995, 11.76901471789268315551874749646

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