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Properties

Label 2-3724-133.18-c0-0-0
Degree $2$
Conductor $3724$
Sign $0.386 - 0.922i$
Analytic cond. $1.85851$
Root an. cond. $1.36327$
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  + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)9-s + (−1 + 1.73i)11-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)23-s − 43-s + (−0.499 + 0.866i)45-s + (1 + 1.73i)47-s + 1.99·55-s + (1 + 1.73i)61-s + (1 − 1.73i)73-s + (−0.499 + 0.866i)81-s + 83-s + 0.999·85-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)9-s + (−1 + 1.73i)11-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)23-s − 43-s + (−0.499 + 0.866i)45-s + (1 + 1.73i)47-s + 1.99·55-s + (1 + 1.73i)61-s + (1 − 1.73i)73-s + (−0.499 + 0.866i)81-s + 83-s + 0.999·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7737212788\)
\(L(\frac12)\) \(\approx\) \(0.7737212788\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.854436274162758773897482999374, −8.024704378023574529486919179705, −7.54159610319677670411404682697, −6.66668156392662870522273020341, −5.75540549233699440635652560274, −5.00119434713844587206790795065, −4.30911460030840641247775451153, −3.50783283502260783260818831289, −2.37397407836075647218468772072, −1.26256569179095795781719619873, 0.46641939136358031040070026561, 2.45281514090957435251667184677, 2.90467932299218865150102715240, 3.70904358864670238223929514549, 5.06624198221128157906175646619, 5.33964955789864025521223483254, 6.48715212698110494694484387156, 7.07799398365350954237948498892, 7.88825671336480404332242423823, 8.457773126562683940816771489070

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