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Properties

Label 2-1620-12.11-c1-0-22
Degree $2$
Conductor $1620$
Sign $0.904 - 0.426i$
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  + (−0.309 − 1.37i)2-s + (−1.80 + 0.853i)4-s + i·5-s + 0.937i·7-s + (1.73 + 2.23i)8-s + (1.37 − 0.309i)10-s + 0.583·11-s + 1.15·13-s + (1.29 − 0.289i)14-s + (2.54 − 3.08i)16-s + 0.238i·17-s − 7.00i·19-s + (−0.853 − 1.80i)20-s + (−0.180 − 0.805i)22-s − 5.06·23-s + ⋯
L(s)  = 1  + (−0.218 − 0.975i)2-s + (−0.904 + 0.426i)4-s + 0.447i·5-s + 0.354i·7-s + (0.614 + 0.789i)8-s + (0.436 − 0.0977i)10-s + 0.176·11-s + 0.321·13-s + (0.345 − 0.0774i)14-s + (0.635 − 0.771i)16-s + 0.0577i·17-s − 1.60i·19-s + (−0.190 − 0.404i)20-s + (−0.0384 − 0.171i)22-s − 1.05·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.904 - 0.426i$
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: \((2,\ 1620,\ (\ :1/2),\ 0.904 - 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.124291100\)
\(L(\frac12)\) \(\approx\) \(1.124291100\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 1.37i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 - 0.937iT - 7T^{2} \)
11 \( 1 - 0.583T + 11T^{2} \)
13 \( 1 - 1.15T + 13T^{2} \)
17 \( 1 - 0.238iT - 17T^{2} \)
19 \( 1 + 7.00iT - 19T^{2} \)
23 \( 1 + 5.06T + 23T^{2} \)
29 \( 1 - 9.14iT - 29T^{2} \)
31 \( 1 - 6.27iT - 31T^{2} \)
37 \( 1 - 3.50T + 37T^{2} \)
41 \( 1 - 3.30iT - 41T^{2} \)
43 \( 1 - 10.7iT - 43T^{2} \)
47 \( 1 - 8.55T + 47T^{2} \)
53 \( 1 - 11.9iT - 53T^{2} \)
59 \( 1 + 9.89T + 59T^{2} \)
61 \( 1 - 6.66T + 61T^{2} \)
67 \( 1 + 8.88iT - 67T^{2} \)
71 \( 1 + 8.95T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 - 5.05iT - 79T^{2} \)
83 \( 1 + 4.72T + 83T^{2} \)
89 \( 1 - 2.36iT - 89T^{2} \)
97 \( 1 - 4.53T + 97T^{2} \)
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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

−9.339849185532065025998216509721, −8.976545445196644071936764306733, −8.046137101143613708917262146601, −7.17191607719134188121897143535, −6.20032671805845887673576506289, −5.12325317872027696253085950419, −4.28283969138412429358223931552, −3.21221629171865033796133745568, −2.48827534966647219800855067447, −1.21464328310497966371666764336, 0.53273807232904568802556892242, 1.94363307947088505591421861852, 3.89072707550453846098428488232, 4.21361255857945755029492333368, 5.62182346131515733032222648075, 5.93690472270978345031992469674, 7.00039081615816082495070923022, 7.893072103298504900181152868190, 8.280239371420479112702982547647, 9.261002933885332781930771846112

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