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Properties

Label 2-1620-12.11-c1-0-19
Degree $2$
Conductor $1620$
Sign $0.0161 - 0.999i$
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  + (−1.41 − 0.0114i)2-s + (1.99 + 0.0323i)4-s + i·5-s − 2.33i·7-s + (−2.82 − 0.0687i)8-s + (0.0114 − 1.41i)10-s + 4.21·11-s − 4.54·13-s + (−0.0266 + 3.29i)14-s + (3.99 + 0.129i)16-s + 7.60i·17-s − 0.719i·19-s + (−0.0323 + 1.99i)20-s + (−5.95 − 0.0482i)22-s − 7.46·23-s + ⋯
L(s)  = 1  + (−0.999 − 0.00810i)2-s + (0.999 + 0.0161i)4-s + 0.447i·5-s − 0.880i·7-s + (−0.999 − 0.0242i)8-s + (0.00362 − 0.447i)10-s + 1.27·11-s − 1.25·13-s + (−0.00713 + 0.880i)14-s + (0.999 + 0.0323i)16-s + 1.84i·17-s − 0.165i·19-s + (−0.00724 + 0.447i)20-s + (−1.27 − 0.0102i)22-s − 1.55·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0161 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0161 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.0161 - 0.999i$
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: \((2,\ 1620,\ (\ :1/2),\ 0.0161 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7696951009\)
\(L(\frac12)\) \(\approx\) \(0.7696951009\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0114i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 + 2.33iT - 7T^{2} \)
11 \( 1 - 4.21T + 11T^{2} \)
13 \( 1 + 4.54T + 13T^{2} \)
17 \( 1 - 7.60iT - 17T^{2} \)
19 \( 1 + 0.719iT - 19T^{2} \)
23 \( 1 + 7.46T + 23T^{2} \)
29 \( 1 - 4.77iT - 29T^{2} \)
31 \( 1 + 4.32iT - 31T^{2} \)
37 \( 1 - 3.84T + 37T^{2} \)
41 \( 1 - 5.69iT - 41T^{2} \)
43 \( 1 - 12.1iT - 43T^{2} \)
47 \( 1 - 9.42T + 47T^{2} \)
53 \( 1 + 5.00iT - 53T^{2} \)
59 \( 1 + 0.00580T + 59T^{2} \)
61 \( 1 + 5.10T + 61T^{2} \)
67 \( 1 - 9.45iT - 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 6.66T + 73T^{2} \)
79 \( 1 + 3.40iT - 79T^{2} \)
83 \( 1 - 4.61T + 83T^{2} \)
89 \( 1 + 3.08iT - 89T^{2} \)
97 \( 1 + 4.11T + 97T^{2} \)
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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

−9.740781693725274442158753426369, −8.843246221295643685270899722154, −7.903188061141268149003202503612, −7.39767163586597316961763861249, −6.46990090459457703697502453433, −5.99693392218231705994072007212, −4.36149743018705274237885902604, −3.59499858130151854282071807366, −2.30377942472573377390374886923, −1.24228108512932827153583629277, 0.43277223579323727952042734999, 1.93516690496922454793230276225, 2.73237767256566685008877842734, 4.10721733081710525022873653824, 5.28889663707908118953679563381, 6.03774694568006128626130496794, 7.03769299259343056360970130913, 7.62706825360534300064217436630, 8.633066567839101114019791710691, 9.261690229457911602353965818777

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