Properties

Label 2-1620-1620.319-c0-0-1
Degree $2$
Conductor $1620$
Sign $0.952 - 0.305i$
Analytic cond. $0.808485$
Root an. cond. $0.899158$
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  + (−0.973 + 0.230i)2-s + (0.993 − 0.116i)3-s + (0.893 − 0.448i)4-s + (0.597 + 0.802i)5-s + (−0.939 + 0.342i)6-s + (0.661 − 0.435i)7-s + (−0.766 + 0.642i)8-s + (0.973 − 0.230i)9-s + (−0.766 − 0.642i)10-s + (0.835 − 0.549i)12-s + (−0.543 + 0.576i)14-s + (0.686 + 0.727i)15-s + (0.597 − 0.802i)16-s + (−0.893 + 0.448i)18-s + (0.893 + 0.448i)20-s + (0.606 − 0.509i)21-s + ⋯
L(s)  = 1  + (−0.973 + 0.230i)2-s + (0.993 − 0.116i)3-s + (0.893 − 0.448i)4-s + (0.597 + 0.802i)5-s + (−0.939 + 0.342i)6-s + (0.661 − 0.435i)7-s + (−0.766 + 0.642i)8-s + (0.973 − 0.230i)9-s + (−0.766 − 0.642i)10-s + (0.835 − 0.549i)12-s + (−0.543 + 0.576i)14-s + (0.686 + 0.727i)15-s + (0.597 − 0.802i)16-s + (−0.893 + 0.448i)18-s + (0.893 + 0.448i)20-s + (0.606 − 0.509i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.952 - 0.305i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ 0.952 - 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.227098243\)
\(L(\frac12)\) \(\approx\) \(1.227098243\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.973 - 0.230i)T \)
3 \( 1 + (-0.993 + 0.116i)T \)
5 \( 1 + (-0.597 - 0.802i)T \)
good7 \( 1 + (-0.661 + 0.435i)T + (0.396 - 0.918i)T^{2} \)
11 \( 1 + (-0.973 - 0.230i)T^{2} \)
13 \( 1 + (0.835 - 0.549i)T^{2} \)
17 \( 1 + (0.939 + 0.342i)T^{2} \)
19 \( 1 + (0.939 - 0.342i)T^{2} \)
23 \( 1 + (1.39 + 0.918i)T + (0.396 + 0.918i)T^{2} \)
29 \( 1 + (-0.0798 - 0.0845i)T + (-0.0581 + 0.998i)T^{2} \)
31 \( 1 + (0.993 + 0.116i)T^{2} \)
37 \( 1 + (-0.766 + 0.642i)T^{2} \)
41 \( 1 + (0.558 + 0.132i)T + (0.893 + 0.448i)T^{2} \)
43 \( 1 + (0.606 + 1.40i)T + (-0.686 + 0.727i)T^{2} \)
47 \( 1 + (-0.113 - 1.94i)T + (-0.993 + 0.116i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.973 + 0.230i)T^{2} \)
61 \( 1 + (1.77 + 0.891i)T + (0.597 + 0.802i)T^{2} \)
67 \( 1 + (0.941 - 0.998i)T + (-0.0581 - 0.998i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (-0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.893 + 0.448i)T^{2} \)
83 \( 1 + (-0.558 + 0.132i)T + (0.893 - 0.448i)T^{2} \)
89 \( 1 + (-1.49 + 1.25i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.286 + 0.957i)T^{2} \)
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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

−9.539399441434884989136648319697, −8.829001158777696013600901478293, −7.987672570997631041142675474716, −7.48817776722494933798613187322, −6.65994177693732402921207605223, −5.93966204170621718523384504253, −4.59136947929876051378942867124, −3.33671790382842719343569462520, −2.34967886936302200791200211434, −1.55560796628950312011126190624, 1.53723745891264633878693134743, 2.13047728166507721372865090792, 3.31139279393079489324273030724, 4.42389452067560972926844224844, 5.49979921506510335766606340057, 6.50106612974386025875082722635, 7.61852726705085711399141373933, 8.206143759367754216173372699646, 8.728067510890130059028148567985, 9.513992469932725967995433269164

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