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Properties

Label 2-3963-3963.80-c0-0-0
Degree $2$
Conductor $3963$
Sign $-0.0463 - 0.998i$
Analytic cond. $1.97779$
Root an. cond. $1.40634$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.841 + 0.540i)3-s − 4-s + (1.75 + 0.125i)7-s + (0.415 − 0.909i)9-s + (0.841 − 0.540i)12-s + (−0.373 + 1.71i)13-s + 16-s + (−1.94 + 0.139i)19-s + (−1.54 + 0.841i)21-s + (0.654 + 0.755i)25-s + (0.142 + 0.989i)27-s + (−1.75 − 0.125i)28-s + (0.755 − 1.65i)31-s + (−0.415 + 0.909i)36-s + (0.540 − 0.158i)37-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)3-s − 4-s + (1.75 + 0.125i)7-s + (0.415 − 0.909i)9-s + (0.841 − 0.540i)12-s + (−0.373 + 1.71i)13-s + 16-s + (−1.94 + 0.139i)19-s + (−1.54 + 0.841i)21-s + (0.654 + 0.755i)25-s + (0.142 + 0.989i)27-s + (−1.75 − 0.125i)28-s + (0.755 − 1.65i)31-s + (−0.415 + 0.909i)36-s + (0.540 − 0.158i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3963 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0463 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3963 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0463 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3963\)    =    \(3 \cdot 1321\)
Sign: $-0.0463 - 0.998i$
Analytic conductor: \(1.97779\)
Root analytic conductor: \(1.40634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3963} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3963,\ (\ :0),\ -0.0463 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8051559932\)
\(L(\frac12)\) \(\approx\) \(0.8051559932\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.841 - 0.540i)T \)
1321 \( 1 + (-0.959 - 0.281i)T \)
good2 \( 1 + T^{2} \)
5 \( 1 + (-0.654 - 0.755i)T^{2} \)
7 \( 1 + (-1.75 - 0.125i)T + (0.989 + 0.142i)T^{2} \)
11 \( 1 + (0.959 + 0.281i)T^{2} \)
13 \( 1 + (0.373 - 1.71i)T + (-0.909 - 0.415i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (1.94 - 0.139i)T + (0.989 - 0.142i)T^{2} \)
23 \( 1 + (-0.540 + 0.841i)T^{2} \)
29 \( 1 + (-0.142 - 0.989i)T^{2} \)
31 \( 1 + (-0.755 + 1.65i)T + (-0.654 - 0.755i)T^{2} \)
37 \( 1 + (-0.540 + 0.158i)T + (0.841 - 0.540i)T^{2} \)
41 \( 1 + (-0.654 - 0.755i)T^{2} \)
43 \( 1 + (-1.27 - 0.817i)T + (0.415 + 0.909i)T^{2} \)
47 \( 1 + (-0.755 - 0.654i)T^{2} \)
53 \( 1 + (0.142 + 0.989i)T^{2} \)
59 \( 1 + (-0.281 + 0.959i)T^{2} \)
61 \( 1 + (0.114 - 0.0855i)T + (0.281 - 0.959i)T^{2} \)
67 \( 1 + (1.64 - 0.898i)T + (0.540 - 0.841i)T^{2} \)
71 \( 1 + (-0.959 - 0.281i)T^{2} \)
73 \( 1 + (1.75 - 0.654i)T + (0.755 - 0.654i)T^{2} \)
79 \( 1 + (-1.40 + 1.05i)T + (0.281 - 0.959i)T^{2} \)
83 \( 1 + (-0.989 - 0.142i)T^{2} \)
89 \( 1 + (0.909 + 0.415i)T^{2} \)
97 \( 1 + (-0.697 - 1.86i)T + (-0.755 + 0.654i)T^{2} \)
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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

−9.088326337579770960155616516766, −8.180936050075892959245826913592, −7.46836512737683367385014702866, −6.41929720120207511714524755680, −5.73672696601128518301441932567, −4.79879594593941038121109294435, −4.40914815135418232795892767512, −4.04131830409881830752364935573, −2.27851302948975573516325397684, −1.21949066111002836587440555221, 0.61245380711700510437801523300, 1.65135220599291797093843828073, 2.81649836680192977982953845314, 4.36931970727756200765694144845, 4.65484340932331551526814330299, 5.38630913254251333124081867102, 6.02749877887146514987704482979, 7.08475183596494297121284187293, 7.909851481834116012365024860993, 8.291742740929014743937684910403

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