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Properties

Label 2-1620-12.11-c1-0-37
Degree $2$
Conductor $1620$
Sign $-0.312 - 0.950i$
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  + (1.14 + 0.829i)2-s + (0.624 + 1.90i)4-s + i·5-s − 2.90i·7-s + (−0.860 + 2.69i)8-s + (−0.829 + 1.14i)10-s + 2.72·11-s − 1.84·13-s + (2.40 − 3.32i)14-s + (−3.22 + 2.37i)16-s + 6.64i·17-s + 4.90i·19-s + (−1.90 + 0.624i)20-s + (3.12 + 2.26i)22-s + 6.63·23-s + ⋯
L(s)  = 1  + (0.809 + 0.586i)2-s + (0.312 + 0.950i)4-s + 0.447i·5-s − 1.09i·7-s + (−0.304 + 0.952i)8-s + (−0.262 + 0.362i)10-s + 0.822·11-s − 0.512·13-s + (0.643 − 0.889i)14-s + (−0.805 + 0.593i)16-s + 1.61i·17-s + 1.12i·19-s + (−0.424 + 0.139i)20-s + (0.666 + 0.482i)22-s + 1.38·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.312 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.312 - 0.950i$
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: \((2,\ 1620,\ (\ :1/2),\ -0.312 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.649633091\)
\(L(\frac12)\) \(\approx\) \(2.649633091\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.14 - 0.829i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 + 2.90iT - 7T^{2} \)
11 \( 1 - 2.72T + 11T^{2} \)
13 \( 1 + 1.84T + 13T^{2} \)
17 \( 1 - 6.64iT - 17T^{2} \)
19 \( 1 - 4.90iT - 19T^{2} \)
23 \( 1 - 6.63T + 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 - 5.22iT - 31T^{2} \)
37 \( 1 + 0.280T + 37T^{2} \)
41 \( 1 - 8.44iT - 41T^{2} \)
43 \( 1 - 7.15iT - 43T^{2} \)
47 \( 1 + 1.48T + 47T^{2} \)
53 \( 1 + 4.62iT - 53T^{2} \)
59 \( 1 - 5.12T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 - 1.71iT - 67T^{2} \)
71 \( 1 - 5.25T + 71T^{2} \)
73 \( 1 + 9.36T + 73T^{2} \)
79 \( 1 + 8.18iT - 79T^{2} \)
83 \( 1 + 9.15T + 83T^{2} \)
89 \( 1 + 14.8iT - 89T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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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

−9.718863903659115766778303830611, −8.512104376248356593261623466943, −7.88771270643422330415841500823, −7.01103804133614400462422052782, −6.50522428836858418616068164677, −5.66077903249764371877191983869, −4.48794647173337402992715317481, −3.88718199403515428499562099348, −3.05012000615004411749100336621, −1.56373034403541457234266746121, 0.810192039120617322020350863785, 2.25198029275338141186598388242, 2.95542051750602636076833863507, 4.14404911676158626911808462887, 5.19393038303585813790125369097, 5.38876853944399531229806315953, 6.73097349292374076345144894999, 7.21135755629160505542697018229, 8.844515598491023177929602384370, 9.124918469101940342832287381671

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