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Properties

Label 2-1620-9.4-c3-0-38
Degree $2$
Conductor $1620$
Sign $-0.766 + 0.642i$
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 + (−8.05 − 13.9i)7-s + (27.7 + 48.0i)11-s + (−21.7 + 37.6i)13-s + 25.5·17-s − 103.·19-s + (2.23 − 3.87i)23-s + (−12.5 − 21.6i)25-s + (2.23 + 3.87i)29-s + (22.5 − 39.0i)31-s − 80.5·35-s + 69.5·37-s + (241. − 418. i)41-s + (−75.8 − 131. i)43-s + (−33.8 − 58.5i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.434 − 0.753i)7-s + (0.760 + 1.31i)11-s + (−0.463 + 0.802i)13-s + 0.364·17-s − 1.24·19-s + (0.0202 − 0.0351i)23-s + (−0.100 − 0.173i)25-s + (0.0143 + 0.0248i)29-s + (0.130 − 0.226i)31-s − 0.388·35-s + 0.309·37-s + (0.920 − 1.59i)41-s + (−0.269 − 0.466i)43-s + (−0.104 − 0.181i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.766 + 0.642i$
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: \((2,\ 1620,\ (\ :3/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7996596744\)
\(L(\frac12)\) \(\approx\) \(0.7996596744\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.05 + 13.9i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-27.7 - 48.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (21.7 - 37.6i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 25.5T + 4.91e3T^{2} \)
19 \( 1 + 103.T + 6.85e3T^{2} \)
23 \( 1 + (-2.23 + 3.87i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-2.23 - 3.87i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-22.5 + 39.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 69.5T + 5.06e4T^{2} \)
41 \( 1 + (-241. + 418. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (75.8 + 131. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (33.8 + 58.5i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 278.T + 1.48e5T^{2} \)
59 \( 1 + (-128. + 223. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-244. - 423. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-386. + 669. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 536.T + 3.57e5T^{2} \)
73 \( 1 + 65.5T + 3.89e5T^{2} \)
79 \( 1 + (374. + 648. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-630. - 1.09e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.32e3T + 7.04e5T^{2} \)
97 \( 1 + (-16.4 - 28.5i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−8.907110883655815803897677092447, −7.84874175515102046292290005845, −6.95782663782570324965954380185, −6.55941482335017124178489916077, −5.37093620961798102707620693991, −4.32014530037584904282880562542, −3.94750271347046415311155274115, −2.39088322661713151520323247834, −1.51101031167891633535949411802, −0.17935480763125213467603313420, 1.14014571374305311618709831808, 2.56974559487338468981060365887, 3.17939559430296957432440393775, 4.26742653957870021174290829372, 5.49422713617261500627009608872, 6.10701127677770113096501931584, 6.71262710310628921600432502851, 7.916165638158765667065435856762, 8.534123517756536216690637270583, 9.349138672570166451368660618027

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