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Properties

Label 2-1620-12.11-c1-0-30
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

\(L(1)\) \(\approx\) \(1.578267611\)
\(L(\frac12)\) \(\approx\) \(1.578267611\)
\(L(\frac{3}{2})\) 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

−9.544654311907253097262875896544, −8.694453508155383408121119472389, −7.894793399368251789545208233582, −6.91141830314031016849924894858, −5.88790014046562486913930908725, −5.12996225819847882937840171731, −3.99456863894084366824714341155, −3.38328917040320907279963352838, −2.27961234545565244581776710686, −1.09803054607052576648254363742, 0.70102768661558215596953970526, 2.56979068608268475488089062504, 3.69973503591433030715221145110, 4.80536710823503496813420915636, 5.22739016400310718626075189620, 6.32941103535513824177813012868, 6.90716294508114352215324373895, 7.917323520699983765223821137752, 8.532486126338957636650435131670, 9.259214418352743977037151159957

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