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Properties

Label 2-392-392.205-c1-0-17
Degree $2$
Conductor $392$
Sign $0.591 - 0.806i$
Analytic cond. $3.13013$
Root an. cond. $1.76921$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.19 − 0.762i)2-s + (−0.209 + 1.39i)3-s + (0.836 + 1.81i)4-s + (1.73 − 0.679i)5-s + (1.31 − 1.49i)6-s + (−0.592 + 2.57i)7-s + (0.389 − 2.80i)8-s + (0.971 + 0.299i)9-s + (−2.58 − 0.511i)10-s + (−0.224 − 0.728i)11-s + (−2.70 + 0.783i)12-s + (1.74 + 0.398i)13-s + (2.67 − 2.61i)14-s + (0.582 + 2.55i)15-s + (−2.60 + 3.03i)16-s + (4.21 + 2.87i)17-s + ⋯
L(s)  = 1  + (−0.842 − 0.539i)2-s + (−0.121 + 0.803i)3-s + (0.418 + 0.908i)4-s + (0.774 − 0.303i)5-s + (0.535 − 0.611i)6-s + (−0.223 + 0.974i)7-s + (0.137 − 0.990i)8-s + (0.323 + 0.0999i)9-s + (−0.815 − 0.161i)10-s + (−0.0677 − 0.219i)11-s + (−0.780 + 0.226i)12-s + (0.483 + 0.110i)13-s + (0.714 − 0.700i)14-s + (0.150 + 0.659i)15-s + (−0.650 + 0.759i)16-s + (1.02 + 0.697i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.591 - 0.806i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ 0.591 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.894829 + 0.453668i\)
\(L(\frac12)\) \(\approx\) \(0.894829 + 0.453668i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 + 0.762i)T \)
7 \( 1 + (0.592 - 2.57i)T \)
good3 \( 1 + (0.209 - 1.39i)T + (-2.86 - 0.884i)T^{2} \)
5 \( 1 + (-1.73 + 0.679i)T + (3.66 - 3.40i)T^{2} \)
11 \( 1 + (0.224 + 0.728i)T + (-9.08 + 6.19i)T^{2} \)
13 \( 1 + (-1.74 - 0.398i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (-4.21 - 2.87i)T + (6.21 + 15.8i)T^{2} \)
19 \( 1 + (7.23 + 4.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.115 - 0.0785i)T + (8.40 - 21.4i)T^{2} \)
29 \( 1 + (-2.76 - 5.74i)T + (-18.0 + 22.6i)T^{2} \)
31 \( 1 + (-3.93 - 6.81i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.03 - 0.302i)T + (36.5 + 5.51i)T^{2} \)
41 \( 1 + (-2.31 + 2.90i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (-6.71 + 5.35i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (-4.15 - 3.85i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (7.02 - 0.526i)T + (52.4 - 7.89i)T^{2} \)
59 \( 1 + (7.74 + 3.03i)T + (43.2 + 40.1i)T^{2} \)
61 \( 1 + (7.47 + 0.559i)T + (60.3 + 9.09i)T^{2} \)
67 \( 1 + (-6.85 + 3.95i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.31 - 1.11i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-0.0710 + 0.0659i)T + (5.45 - 72.7i)T^{2} \)
79 \( 1 + (2.79 - 4.84i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.51 + 2.17i)T + (74.7 - 36.0i)T^{2} \)
89 \( 1 + (0.741 + 0.228i)T + (73.5 + 50.1i)T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−10.97635563492406756416653573332, −10.55105168670314744059697724709, −9.550560454621005295111159740821, −9.004663313535094650645186675292, −8.187133118466248652551518711189, −6.70269731461709644850447726498, −5.67847076357748437579379331146, −4.40065472081456922864650716554, −3.06042347492544007118475289949, −1.69578402138929601517009304343, 0.963507012362228674191529799460, 2.28517749118359507128185713897, 4.31651849007946742435227110507, 6.15157816190200477069250577807, 6.29726165461507487688741154755, 7.56809964265498397334140332094, 8.019139070678468607852075116639, 9.563084494615403755335753237781, 10.06271743632664919205128385774, 10.84513998904672491337664916354

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