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Properties

Label 2-3963-3963.2267-c0-0-0
Degree $2$
Conductor $3963$
Sign $-0.421 + 0.907i$
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  + (−0.415 − 0.909i)3-s − 4-s + (−0.340 − 0.254i)7-s + (−0.654 + 0.755i)9-s + (0.415 + 0.909i)12-s + (0.398 + 0.148i)13-s + 16-s + (0.559 − 0.418i)19-s + (−0.0903 + 0.415i)21-s + (0.142 + 0.989i)25-s + (0.959 + 0.281i)27-s + (0.340 + 0.254i)28-s + (0.989 − 1.14i)31-s + (0.654 − 0.755i)36-s + (−0.909 − 0.584i)37-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)3-s − 4-s + (−0.340 − 0.254i)7-s + (−0.654 + 0.755i)9-s + (0.415 + 0.909i)12-s + (0.398 + 0.148i)13-s + 16-s + (0.559 − 0.418i)19-s + (−0.0903 + 0.415i)21-s + (0.142 + 0.989i)25-s + (0.959 + 0.281i)27-s + (0.340 + 0.254i)28-s + (0.989 − 1.14i)31-s + (0.654 − 0.755i)36-s + (−0.909 − 0.584i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3963 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3963 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3963\)    =    \(3 \cdot 1321\)
Sign: $-0.421 + 0.907i$
Analytic conductor: \(1.97779\)
Root analytic conductor: \(1.40634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3963} (2267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3963,\ (\ :0),\ -0.421 + 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6691388785\)
\(L(\frac12)\) \(\approx\) \(0.6691388785\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.415 + 0.909i)T \)
1321 \( 1 + (0.841 - 0.540i)T \)
good2 \( 1 + T^{2} \)
5 \( 1 + (-0.142 - 0.989i)T^{2} \)
7 \( 1 + (0.340 + 0.254i)T + (0.281 + 0.959i)T^{2} \)
11 \( 1 + (-0.841 + 0.540i)T^{2} \)
13 \( 1 + (-0.398 - 0.148i)T + (0.755 + 0.654i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.559 + 0.418i)T + (0.281 - 0.959i)T^{2} \)
23 \( 1 + (0.909 + 0.415i)T^{2} \)
29 \( 1 + (-0.959 - 0.281i)T^{2} \)
31 \( 1 + (-0.989 + 1.14i)T + (-0.142 - 0.989i)T^{2} \)
37 \( 1 + (0.909 + 0.584i)T + (0.415 + 0.909i)T^{2} \)
41 \( 1 + (-0.142 - 0.989i)T^{2} \)
43 \( 1 + (-0.822 + 1.80i)T + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + (-0.989 - 0.142i)T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (-0.540 - 0.841i)T^{2} \)
61 \( 1 + (1.05 + 0.574i)T + (0.540 + 0.841i)T^{2} \)
67 \( 1 + (0.424 - 1.94i)T + (-0.909 - 0.415i)T^{2} \)
71 \( 1 + (0.841 - 0.540i)T^{2} \)
73 \( 1 + (1.98 - 0.142i)T + (0.989 - 0.142i)T^{2} \)
79 \( 1 + (0.373 + 0.203i)T + (0.540 + 0.841i)T^{2} \)
83 \( 1 + (-0.281 - 0.959i)T^{2} \)
89 \( 1 + (-0.755 - 0.654i)T^{2} \)
97 \( 1 + (0.114 + 1.59i)T + (-0.989 + 0.142i)T^{2} \)
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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

−8.535756778153446525484190271918, −7.46491780998185773355978404402, −7.15373839093866872294249889374, −6.05187660280703485998959634668, −5.55611414205706975097572000083, −4.71413874953685783241003783654, −3.82147813823450780135956317901, −2.90060353825739247617192271493, −1.62595765521125052568564008738, −0.50246796065351484750999959624, 1.11275783070350294512291897209, 2.95338015417401444412472492934, 3.50799822036060880386433928179, 4.57719030415120367297163748330, 4.84976336986423067429126940281, 5.96002378533791228950681062218, 6.25439005847905836116256778700, 7.55415633136618704629655660178, 8.424369728082396659373998077755, 8.906176117042699421587894586041

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