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Properties

Label 2-1620-12.11-c1-0-18
Degree $2$
Conductor $1620$
Sign $-0.999 - 0.0196i$
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  + (0.990 + 1.00i)2-s + (−0.0393 + 1.99i)4-s − i·5-s + 2.09i·7-s + (−2.05 + 1.94i)8-s + (1.00 − 0.990i)10-s − 1.30·11-s − 0.549·13-s + (−2.11 + 2.07i)14-s + (−3.99 − 0.157i)16-s + 4.51i·17-s − 3.08i·19-s + (1.99 + 0.0393i)20-s + (−1.29 − 1.32i)22-s − 7.68·23-s + ⋯
L(s)  = 1  + (0.700 + 0.714i)2-s + (−0.0196 + 0.999i)4-s − 0.447i·5-s + 0.791i·7-s + (−0.727 + 0.685i)8-s + (0.319 − 0.313i)10-s − 0.394·11-s − 0.152·13-s + (−0.565 + 0.554i)14-s + (−0.999 − 0.0392i)16-s + 1.09i·17-s − 0.706i·19-s + (0.447 + 0.00878i)20-s + (−0.275 − 0.281i)22-s − 1.60·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.999 - 0.0196i$
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: \((2,\ 1620,\ (\ :1/2),\ -0.999 - 0.0196i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.467197466\)
\(L(\frac12)\) \(\approx\) \(1.467197466\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.990 - 1.00i)T \)
3 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 - 2.09iT - 7T^{2} \)
11 \( 1 + 1.30T + 11T^{2} \)
13 \( 1 + 0.549T + 13T^{2} \)
17 \( 1 - 4.51iT - 17T^{2} \)
19 \( 1 + 3.08iT - 19T^{2} \)
23 \( 1 + 7.68T + 23T^{2} \)
29 \( 1 - 9.43iT - 29T^{2} \)
31 \( 1 - 5.73iT - 31T^{2} \)
37 \( 1 - 2.99T + 37T^{2} \)
41 \( 1 + 4.90iT - 41T^{2} \)
43 \( 1 - 4.16iT - 43T^{2} \)
47 \( 1 + 8.88T + 47T^{2} \)
53 \( 1 + 8.85iT - 53T^{2} \)
59 \( 1 - 2.03T + 59T^{2} \)
61 \( 1 - 1.45T + 61T^{2} \)
67 \( 1 - 6.94iT - 67T^{2} \)
71 \( 1 - 0.468T + 71T^{2} \)
73 \( 1 + 6.61T + 73T^{2} \)
79 \( 1 - 13.4iT - 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + 8.03iT - 89T^{2} \)
97 \( 1 - 15.3T + 97T^{2} \)
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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

−9.590270314232171057897864671671, −8.572345639440376414892977262541, −8.363467447662088567998295641551, −7.29413324912393447730549853865, −6.44907909219322980382818018647, −5.62541335321161959641395982851, −5.02854145690123499274793708722, −4.07217050183259662521272503755, −3.06712844614016165641863974816, −1.93975713324508517302444565244, 0.41298870104952747377043679144, 1.96928471353304535454394784222, 2.90040747485831848729104950784, 3.95107012377221700389197658016, 4.54905986814422324413608226497, 5.72499674801264522753631388648, 6.32665468369844095322210089327, 7.40139760465860238484078114174, 8.031875501618318593840375485885, 9.420718526065291364744810927583

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