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Properties

Label 2-392-56.53-c1-0-33
Degree $2$
Conductor $392$
Sign $-0.597 + 0.801i$
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  + (0.804 − 1.16i)2-s + (−0.591 − 0.341i)3-s + (−0.706 − 1.87i)4-s + (2.80 − 1.61i)5-s + (−0.872 + 0.413i)6-s + (−2.74 − 0.682i)8-s + (−1.26 − 2.19i)9-s + (0.371 − 4.56i)10-s + (2.08 + 1.20i)11-s + (−0.220 + 1.34i)12-s + 3.09i·13-s − 2.21·15-s + (−3.00 + 2.64i)16-s + (1.97 − 3.42i)17-s + (−3.57 − 0.290i)18-s + (−2.33 + 1.35i)19-s + ⋯
L(s)  = 1  + (0.568 − 0.822i)2-s + (−0.341 − 0.197i)3-s + (−0.353 − 0.935i)4-s + (1.25 − 0.724i)5-s + (−0.356 + 0.168i)6-s + (−0.970 − 0.241i)8-s + (−0.422 − 0.731i)9-s + (0.117 − 1.44i)10-s + (0.629 + 0.363i)11-s + (−0.0637 + 0.388i)12-s + 0.858i·13-s − 0.570·15-s + (−0.750 + 0.661i)16-s + (0.479 − 0.830i)17-s + (−0.841 − 0.0685i)18-s + (−0.536 + 0.309i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.597 + 0.801i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ -0.597 + 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.810044 - 1.61420i\)
\(L(\frac12)\) \(\approx\) \(0.810044 - 1.61420i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.804 + 1.16i)T \)
7 \( 1 \)
good3 \( 1 + (0.591 + 0.341i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.80 + 1.61i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.08 - 1.20i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.09iT - 13T^{2} \)
17 \( 1 + (-1.97 + 3.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.33 - 1.35i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.37 + 2.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.01iT - 29T^{2} \)
31 \( 1 + (1.10 - 1.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.30 + 2.48i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.11T + 41T^{2} \)
43 \( 1 - 11.5iT - 43T^{2} \)
47 \( 1 + (3.31 + 5.74i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.23 - 1.29i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.6 - 6.13i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.54 + 4.35i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.01 - 2.89i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.64T + 71T^{2} \)
73 \( 1 + (4.77 - 8.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.838 + 1.45i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.47iT - 83T^{2} \)
89 \( 1 + (-6.98 - 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.37T + 97T^{2} \)
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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

−11.25838953878099891983151991064, −9.929422332652364241455195099282, −9.472670717676492722602639745129, −8.675111542711393160629181556892, −6.71851956242846017445109847763, −6.00256525293487634645606782350, −5.10300525405990702595234230947, −3.99582622220231870754059804808, −2.37982241849163041350219624337, −1.15128204683826256626710093442, 2.40447083738771591864850017800, 3.69726590326234237593591764083, 5.20789878295036531795325185963, 5.87084114234126653581604965972, 6.56692129017961339315498053451, 7.77823556229038677874842452251, 8.725505381993575913047373769877, 9.871827435291331162748380996644, 10.69173083404046713272869297387, 11.60927249377957074168858464758

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