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Properties

Label 2-3963-3963.773-c0-0-0
Degree $2$
Conductor $3963$
Sign $-0.852 + 0.523i$
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  + (0.142 + 0.989i)3-s − 4-s + (0.139 + 0.0303i)7-s + (−0.959 + 0.281i)9-s + (−0.142 − 0.989i)12-s + (0.0855 − 0.114i)13-s + 16-s + (−1.56 + 0.340i)19-s + (−0.0101 + 0.142i)21-s + (−0.841 + 0.540i)25-s + (−0.415 − 0.909i)27-s + (−0.139 − 0.0303i)28-s + (−0.540 + 0.158i)31-s + (0.959 − 0.281i)36-s + (−0.989 + 1.14i)37-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)3-s − 4-s + (0.139 + 0.0303i)7-s + (−0.959 + 0.281i)9-s + (−0.142 − 0.989i)12-s + (0.0855 − 0.114i)13-s + 16-s + (−1.56 + 0.340i)19-s + (−0.0101 + 0.142i)21-s + (−0.841 + 0.540i)25-s + (−0.415 − 0.909i)27-s + (−0.139 − 0.0303i)28-s + (−0.540 + 0.158i)31-s + (0.959 − 0.281i)36-s + (−0.989 + 1.14i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3963 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3963 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3963\)    =    \(3 \cdot 1321\)
Sign: $-0.852 + 0.523i$
Analytic conductor: \(1.97779\)
Root analytic conductor: \(1.40634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3963} (773, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3963,\ (\ :0),\ -0.852 + 0.523i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2126260927\)
\(L(\frac12)\) \(\approx\) \(0.2126260927\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.142 - 0.989i)T \)
1321 \( 1 + (-0.654 - 0.755i)T \)
good2 \( 1 + T^{2} \)
5 \( 1 + (0.841 - 0.540i)T^{2} \)
7 \( 1 + (-0.139 - 0.0303i)T + (0.909 + 0.415i)T^{2} \)
11 \( 1 + (0.654 + 0.755i)T^{2} \)
13 \( 1 + (-0.0855 + 0.114i)T + (-0.281 - 0.959i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (1.56 - 0.340i)T + (0.909 - 0.415i)T^{2} \)
23 \( 1 + (0.989 + 0.142i)T^{2} \)
29 \( 1 + (0.415 + 0.909i)T^{2} \)
31 \( 1 + (0.540 - 0.158i)T + (0.841 - 0.540i)T^{2} \)
37 \( 1 + (0.989 - 1.14i)T + (-0.142 - 0.989i)T^{2} \)
41 \( 1 + (0.841 - 0.540i)T^{2} \)
43 \( 1 + (-0.153 + 1.07i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (0.540 - 0.841i)T^{2} \)
53 \( 1 + (-0.415 - 0.909i)T^{2} \)
59 \( 1 + (0.755 - 0.654i)T^{2} \)
61 \( 1 + (0.148 + 0.398i)T + (-0.755 + 0.654i)T^{2} \)
67 \( 1 + (0.0683 - 0.956i)T + (-0.989 - 0.142i)T^{2} \)
71 \( 1 + (-0.654 - 0.755i)T^{2} \)
73 \( 1 + (0.459 - 0.841i)T + (-0.540 - 0.841i)T^{2} \)
79 \( 1 + (0.0498 + 0.133i)T + (-0.755 + 0.654i)T^{2} \)
83 \( 1 + (-0.909 - 0.415i)T^{2} \)
89 \( 1 + (0.281 + 0.959i)T^{2} \)
97 \( 1 + (1.71 + 0.936i)T + (0.540 + 0.841i)T^{2} \)
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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

−8.998735395543240644613898622415, −8.477627161004467531876922658599, −7.981444939578467374366286755526, −6.83153854600825821055695586469, −5.79234679760204099672428199526, −5.27772870039052624592617738079, −4.41400081368308073325905765557, −3.89573679760379961757140912943, −3.09086946967075101653148371822, −1.80361609688262769482547434526, 0.11755872702090380647748151824, 1.53082987365252776714682383299, 2.48188908880203052281021467429, 3.61287886913129860711201895497, 4.36244604003117274381088604313, 5.26725417404246289402797348479, 6.07615023603585439910912929364, 6.68636999633346841046964491184, 7.71541345150884269498732141523, 8.103089933405437498166693054509

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