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Properties

Label 2-1620-12.11-c1-0-15
Degree $2$
Conductor $1620$
Sign $0.738 - 0.674i$
Analytic cond. $12.9357$
Root an. cond. $3.59663$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.570 − 1.29i)2-s + (−1.34 − 1.47i)4-s − i·5-s + 3.80i·7-s + (−2.67 + 0.904i)8-s + (−1.29 − 0.570i)10-s − 0.568·11-s − 3.24·13-s + (4.92 + 2.17i)14-s + (−0.357 + 3.98i)16-s − 3.81i·17-s − 0.342i·19-s + (−1.47 + 1.34i)20-s + (−0.323 + 0.735i)22-s + 0.230·23-s + ⋯
L(s)  = 1  + (0.403 − 0.915i)2-s + (−0.674 − 0.738i)4-s − 0.447i·5-s + 1.43i·7-s + (−0.947 + 0.319i)8-s + (−0.409 − 0.180i)10-s − 0.171·11-s − 0.900·13-s + (1.31 + 0.580i)14-s + (−0.0894 + 0.995i)16-s − 0.924i·17-s − 0.0786i·19-s + (−0.330 + 0.301i)20-s + (−0.0690 + 0.156i)22-s + 0.0481·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.738 - 0.674i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ 0.738 - 0.674i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.023054135\)
\(L(\frac12)\) \(\approx\) \(1.023054135\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.570 + 1.29i)T \)
3 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 - 3.80iT - 7T^{2} \)
11 \( 1 + 0.568T + 11T^{2} \)
13 \( 1 + 3.24T + 13T^{2} \)
17 \( 1 + 3.81iT - 17T^{2} \)
19 \( 1 + 0.342iT - 19T^{2} \)
23 \( 1 - 0.230T + 23T^{2} \)
29 \( 1 - 7.77iT - 29T^{2} \)
31 \( 1 - 10.0iT - 31T^{2} \)
37 \( 1 + 0.446T + 37T^{2} \)
41 \( 1 - 6.75iT - 41T^{2} \)
43 \( 1 - 9.23iT - 43T^{2} \)
47 \( 1 + 2.07T + 47T^{2} \)
53 \( 1 - 1.79iT - 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 - 8.78iT - 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 5.29T + 73T^{2} \)
79 \( 1 + 13.2iT - 79T^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 + 3.79iT - 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−9.428339804234926663223097405776, −8.977188341966469772761528807254, −8.222263952673820329317282412472, −6.98469167310737293205630023210, −5.92490558082215510502817201256, −5.05900120275645236127098710366, −4.75694070857610870715975353403, −3.19666906391732943776664844207, −2.59019068445386381353496838866, −1.43074435024690903956242245659, 0.34440050197358451031890576256, 2.36929242567902221314034509952, 3.76411610317760171336826292289, 4.15475706624693644623507579639, 5.25045891252111881340990950365, 6.18763436041206704865926521498, 6.92800320217055724636325621428, 7.64526342982916841687926058009, 8.068402269254747297404519263072, 9.286644975238629728448615776396

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