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Properties

Label 2-3963-3963.3908-c0-0-0
Degree $2$
Conductor $3963$
Sign $-0.0911 - 0.995i$
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  + (0.959 + 0.281i)3-s − 4-s + (−0.418 + 1.12i)7-s + (0.841 + 0.540i)9-s + (−0.959 − 0.281i)12-s + (−0.574 − 1.05i)13-s + 16-s + (0.613 + 1.64i)19-s + (−0.718 + 0.959i)21-s + (−0.415 − 0.909i)25-s + (0.654 + 0.755i)27-s + (0.418 − 1.12i)28-s + (0.909 + 0.584i)31-s + (−0.841 − 0.540i)36-s + (−0.281 + 1.95i)37-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)3-s − 4-s + (−0.418 + 1.12i)7-s + (0.841 + 0.540i)9-s + (−0.959 − 0.281i)12-s + (−0.574 − 1.05i)13-s + 16-s + (0.613 + 1.64i)19-s + (−0.718 + 0.959i)21-s + (−0.415 − 0.909i)25-s + (0.654 + 0.755i)27-s + (0.418 − 1.12i)28-s + (0.909 + 0.584i)31-s + (−0.841 − 0.540i)36-s + (−0.281 + 1.95i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3963 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0911 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3963 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0911 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3963\)    =    \(3 \cdot 1321\)
Sign: $-0.0911 - 0.995i$
Analytic conductor: \(1.97779\)
Root analytic conductor: \(1.40634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3963} (3908, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3963,\ (\ :0),\ -0.0911 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.216191353\)
\(L(\frac12)\) \(\approx\) \(1.216191353\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.959 - 0.281i)T \)
1321 \( 1 + (-0.142 - 0.989i)T \)
good2 \( 1 + T^{2} \)
5 \( 1 + (0.415 + 0.909i)T^{2} \)
7 \( 1 + (0.418 - 1.12i)T + (-0.755 - 0.654i)T^{2} \)
11 \( 1 + (0.142 + 0.989i)T^{2} \)
13 \( 1 + (0.574 + 1.05i)T + (-0.540 + 0.841i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (-0.613 - 1.64i)T + (-0.755 + 0.654i)T^{2} \)
23 \( 1 + (0.281 + 0.959i)T^{2} \)
29 \( 1 + (-0.654 - 0.755i)T^{2} \)
31 \( 1 + (-0.909 - 0.584i)T + (0.415 + 0.909i)T^{2} \)
37 \( 1 + (0.281 - 1.95i)T + (-0.959 - 0.281i)T^{2} \)
41 \( 1 + (0.415 + 0.909i)T^{2} \)
43 \( 1 + (1.74 - 0.512i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 + (-0.909 - 0.415i)T^{2} \)
53 \( 1 + (0.654 + 0.755i)T^{2} \)
59 \( 1 + (0.989 - 0.142i)T^{2} \)
61 \( 1 + (-0.133 - 1.86i)T + (-0.989 + 0.142i)T^{2} \)
67 \( 1 + (-1.17 + 1.56i)T + (-0.281 - 0.959i)T^{2} \)
71 \( 1 + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (1.90 - 0.415i)T + (0.909 - 0.415i)T^{2} \)
79 \( 1 + (-0.0855 - 1.19i)T + (-0.989 + 0.142i)T^{2} \)
83 \( 1 + (0.755 + 0.654i)T^{2} \)
89 \( 1 + (0.540 - 0.841i)T^{2} \)
97 \( 1 + (0.148 + 0.682i)T + (-0.909 + 0.415i)T^{2} \)
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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

−8.684233085430796186485122232155, −8.222633735764497221526813961780, −7.82659853693476720027239793546, −6.59871216359333063938012116677, −5.65612364080071987625467372353, −5.06934259800450119920984333938, −4.22897689547082575499278768742, −3.24898318512535380957546674199, −2.81221324944144587401649434896, −1.49165566002822122230051056124, 0.64787816535622330747048983252, 1.93386240308179761753878565427, 3.10665816653682007147269935610, 3.84703675183392973499469438145, 4.45219726002443527003782976836, 5.22706571010864561327815658814, 6.51429864348023006658388193653, 7.20954724474643041570842057923, 7.58658218819191358449118267523, 8.600939684699249616336360827716

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