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Properties

Label 2-1620-12.11-c1-0-38
Degree $2$
Conductor $1620$
Sign $0.900 + 0.434i$
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  + (−1.37 − 0.315i)2-s + (1.80 + 0.869i)4-s + i·5-s − 2.19i·7-s + (−2.20 − 1.76i)8-s + (0.315 − 1.37i)10-s − 4.82·11-s − 0.728·13-s + (−0.692 + 3.02i)14-s + (2.48 + 3.13i)16-s + 6.46i·17-s − 3.21i·19-s + (−0.869 + 1.80i)20-s + (6.65 + 1.52i)22-s + 7.34·23-s + ⋯
L(s)  = 1  + (−0.974 − 0.223i)2-s + (0.900 + 0.434i)4-s + 0.447i·5-s − 0.829i·7-s + (−0.780 − 0.624i)8-s + (0.0997 − 0.435i)10-s − 1.45·11-s − 0.202·13-s + (−0.185 + 0.808i)14-s + (0.621 + 0.783i)16-s + 1.56i·17-s − 0.737i·19-s + (−0.194 + 0.402i)20-s + (1.41 + 0.324i)22-s + 1.53·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.900 + 0.434i$
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: \((2,\ 1620,\ (\ :1/2),\ 0.900 + 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9139141803\)
\(L(\frac12)\) \(\approx\) \(0.9139141803\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.315i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 + 2.19iT - 7T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 + 0.728T + 13T^{2} \)
17 \( 1 - 6.46iT - 17T^{2} \)
19 \( 1 + 3.21iT - 19T^{2} \)
23 \( 1 - 7.34T + 23T^{2} \)
29 \( 1 + 0.275iT - 29T^{2} \)
31 \( 1 - 0.919iT - 31T^{2} \)
37 \( 1 - 8.84T + 37T^{2} \)
41 \( 1 + 10.0iT - 41T^{2} \)
43 \( 1 + 1.68iT - 43T^{2} \)
47 \( 1 - 6.79T + 47T^{2} \)
53 \( 1 - 0.913iT - 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 - 8.55T + 61T^{2} \)
67 \( 1 + 1.72iT - 67T^{2} \)
71 \( 1 - 4.71T + 71T^{2} \)
73 \( 1 - 8.12T + 73T^{2} \)
79 \( 1 + 2.12iT - 79T^{2} \)
83 \( 1 - 7.09T + 83T^{2} \)
89 \( 1 - 1.54iT - 89T^{2} \)
97 \( 1 - 6.15T + 97T^{2} \)
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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

−9.368531822271958515349573770162, −8.537668696545072740339560394286, −7.71450109641015589768466441606, −7.22598012904278442983512302627, −6.38254546150821860388094857156, −5.34620435539390972315090284608, −4.08386683671442170981273004611, −3.05486543835348159447632962384, −2.17134642446184045660375092860, −0.69016003118060767053471172177, 0.827940820302489402188163411093, 2.36447415149241156930448416679, 2.96835035415425941177925580446, 4.85843126461845106809972512507, 5.38913767862611145799741506655, 6.27482048808319026215085854389, 7.37131921742884593903707198445, 7.88323458713374225233051130736, 8.702858699532652944915980296931, 9.413849856282509392465752841491

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