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Properties

Label 2-3332-3332.543-c0-0-3
Degree $2$
Conductor $3332$
Sign $0.788 + 0.615i$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
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  + (−0.826 − 0.563i)2-s + (0.997 + 0.925i)3-s + (0.365 + 0.930i)4-s + (−0.302 − 1.32i)6-s + (−0.974 − 0.222i)7-s + (0.222 − 0.974i)8-s + (0.0635 + 0.848i)9-s + (0.0841 − 1.12i)11-s + (−0.496 + 1.26i)12-s + (1.48 − 0.716i)13-s + (0.680 + 0.733i)14-s + (−0.733 + 0.680i)16-s + (−0.988 + 0.149i)17-s + (0.425 − 0.736i)18-s + (−0.766 − 1.12i)21-s + (−0.702 + 0.880i)22-s + ⋯
L(s)  = 1  + (−0.826 − 0.563i)2-s + (0.997 + 0.925i)3-s + (0.365 + 0.930i)4-s + (−0.302 − 1.32i)6-s + (−0.974 − 0.222i)7-s + (0.222 − 0.974i)8-s + (0.0635 + 0.848i)9-s + (0.0841 − 1.12i)11-s + (−0.496 + 1.26i)12-s + (1.48 − 0.716i)13-s + (0.680 + 0.733i)14-s + (−0.733 + 0.680i)16-s + (−0.988 + 0.149i)17-s + (0.425 − 0.736i)18-s + (−0.766 − 1.12i)21-s + (−0.702 + 0.880i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.788 + 0.615i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.788 + 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.084405823\)
\(L(\frac12)\) \(\approx\) \(1.084405823\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.826 + 0.563i)T \)
7 \( 1 + (0.974 + 0.222i)T \)
17 \( 1 + (0.988 - 0.149i)T \)
good3 \( 1 + (-0.997 - 0.925i)T + (0.0747 + 0.997i)T^{2} \)
5 \( 1 + (-0.826 + 0.563i)T^{2} \)
11 \( 1 + (-0.0841 + 1.12i)T + (-0.988 - 0.149i)T^{2} \)
13 \( 1 + (-1.48 + 0.716i)T + (0.623 - 0.781i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.858 + 0.129i)T + (0.955 + 0.294i)T^{2} \)
29 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 + (-0.974 + 1.68i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.733 + 0.680i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (-0.365 - 0.930i)T^{2} \)
53 \( 1 + (-0.698 - 1.77i)T + (-0.733 + 0.680i)T^{2} \)
59 \( 1 + (-0.826 - 0.563i)T^{2} \)
61 \( 1 + (0.733 + 0.680i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.367 + 0.460i)T + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.365 + 0.930i)T^{2} \)
79 \( 1 + (0.997 + 1.72i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.109 - 1.46i)T + (-0.988 + 0.149i)T^{2} \)
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

−8.742294386623724572680217404746, −8.421067427998082022721063648268, −7.65554606257011964193984379702, −6.39877422783154697418861513609, −6.04786339315557409410555776879, −4.30654694241900634250566130921, −3.81269549821643372520611551279, −3.09778967226509103587018160304, −2.47916111899799841675566680203, −0.812594350950895735525040373257, 1.35483410091838821716398889191, 2.11194521950529074064507313042, 3.03328810493551691038282577644, 4.16858002937260766617424397861, 5.32651785516326515500264831298, 6.50769190276427528181751078812, 6.72050677911481539877920795359, 7.30313566963934862664605635849, 8.430785047807776942931765474445, 8.601775688337283700608350696991

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