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Properties

Label 2-1620-12.11-c1-0-4
Degree $2$
Conductor $1620$
Sign $-0.750 + 0.660i$
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  + (0.499 + 1.32i)2-s + (−1.50 + 1.32i)4-s − i·5-s + 1.20i·7-s + (−2.49 − 1.32i)8-s + (1.32 − 0.499i)10-s + 3.04·11-s − 5.07·13-s + (−1.59 + 0.603i)14-s + (0.505 − 3.96i)16-s + 2.23i·17-s + 2.53i·19-s + (1.32 + 1.50i)20-s + (1.52 + 4.02i)22-s − 6.96·23-s + ⋯
L(s)  = 1  + (0.353 + 0.935i)2-s + (−0.750 + 0.660i)4-s − 0.447i·5-s + 0.457i·7-s + (−0.883 − 0.468i)8-s + (0.418 − 0.157i)10-s + 0.917·11-s − 1.40·13-s + (−0.427 + 0.161i)14-s + (0.126 − 0.991i)16-s + 0.543i·17-s + 0.582i·19-s + (0.295 + 0.335i)20-s + (0.324 + 0.858i)22-s − 1.45·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 + 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.750 + 0.660i$
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: \((2,\ 1620,\ (\ :1/2),\ -0.750 + 0.660i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5417716619\)
\(L(\frac12)\) \(\approx\) \(0.5417716619\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.499 - 1.32i)T \)
3 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 - 1.20iT - 7T^{2} \)
11 \( 1 - 3.04T + 11T^{2} \)
13 \( 1 + 5.07T + 13T^{2} \)
17 \( 1 - 2.23iT - 17T^{2} \)
19 \( 1 - 2.53iT - 19T^{2} \)
23 \( 1 + 6.96T + 23T^{2} \)
29 \( 1 - 5.59iT - 29T^{2} \)
31 \( 1 + 3.27iT - 31T^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 - 4.27iT - 41T^{2} \)
43 \( 1 + 4.31iT - 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 - 9.42iT - 53T^{2} \)
59 \( 1 + 8.92T + 59T^{2} \)
61 \( 1 + 2.65T + 61T^{2} \)
67 \( 1 + 14.4iT - 67T^{2} \)
71 \( 1 + 3.49T + 71T^{2} \)
73 \( 1 - 5.10T + 73T^{2} \)
79 \( 1 + 3.76iT - 79T^{2} \)
83 \( 1 + 0.572T + 83T^{2} \)
89 \( 1 - 2.40iT - 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−9.578254847454703278972691826786, −8.989749777459482662095787954915, −8.183553466200391098963558730909, −7.49919948864543261188718372011, −6.58214689377160174634452649503, −5.85578698583184996809296224035, −5.03396199092975578776779138725, −4.23794654718244818762073244540, −3.30602005953091471270913162835, −1.84384239609802250374484717395, 0.17579476044803106376367162976, 1.76212638542841151143397646969, 2.73824206672947150876417792797, 3.74372616522912918692566737655, 4.52760277784079163137844112108, 5.39197404527768821438987997043, 6.47661850563887341054380653569, 7.19376093799054448149352871852, 8.231608110261510563526119422567, 9.229748219880316150633848245349

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