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Properties

Label 2-1620-9.4-c3-0-12
Degree $2$
Conductor $1620$
Sign $-0.173 - 0.984i$
Analytic cond. $95.5830$
Root an. cond. $9.77666$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 − 4.33i)5-s + (7.37 + 12.7i)7-s + (−3.62 − 6.28i)11-s + (−30.8 + 53.4i)13-s + 108.·17-s − 56.2·19-s + (23.4 − 40.6i)23-s + (−12.5 − 21.6i)25-s + (107. + 185. i)29-s + (130. − 226. i)31-s + 73.7·35-s − 286·37-s + (−127. + 221. i)41-s + (180. + 312. i)43-s + (2.76 + 4.79i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.398 + 0.689i)7-s + (−0.0993 − 0.172i)11-s + (−0.658 + 1.14i)13-s + 1.54·17-s − 0.679·19-s + (0.212 − 0.368i)23-s + (−0.100 − 0.173i)25-s + (0.685 + 1.18i)29-s + (0.758 − 1.31i)31-s + 0.356·35-s − 1.27·37-s + (−0.486 + 0.843i)41-s + (0.640 + 1.10i)43-s + (0.00858 + 0.0148i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(95.5830\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.707057537\)
\(L(\frac12)\) \(\approx\) \(1.707057537\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-2.5 + 4.33i)T \)
good7 \( 1 + (-7.37 - 12.7i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (3.62 + 6.28i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (30.8 - 53.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 108.T + 4.91e3T^{2} \)
19 \( 1 + 56.2T + 6.85e3T^{2} \)
23 \( 1 + (-23.4 + 40.6i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-107. - 185. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-130. + 226. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 286T + 5.06e4T^{2} \)
41 \( 1 + (127. - 221. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-180. - 312. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-2.76 - 4.79i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 595.T + 1.48e5T^{2} \)
59 \( 1 + (157. - 273. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (138. + 239. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (58.7 - 101. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 192.T + 3.57e5T^{2} \)
73 \( 1 + 756.T + 3.89e5T^{2} \)
79 \( 1 + (-575. - 996. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (70.9 + 122. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 719.T + 7.04e5T^{2} \)
97 \( 1 + (-521. - 903. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.251831409605676563891214195639, −8.465712137899616430459303879329, −7.81726662355970569995390404312, −6.77649809510867505553657432096, −5.97619077566143221982688555557, −5.07797127035801661580730154176, −4.45269285532658154796385342540, −3.16981440131853963063145790687, −2.15302812372449215309544777065, −1.16827059948935085039766567475, 0.38395931992111881053786638959, 1.53723064619384326550493000452, 2.78897217981501579243489035426, 3.59576097136988461831931464739, 4.75407418251024990342193992936, 5.46545630247867720879691782311, 6.41362261966765441533972196262, 7.39775264712800103846577716775, 7.82522602938374365560444024467, 8.729153295095924060195355416794

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