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Properties

Label 2-1620-9.7-c3-0-31
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 + (11 − 19.0i)7-s + (−4.5 + 7.79i)11-s + (−8.5 − 14.7i)13-s + 75·17-s − 4·19-s + (91.5 + 158. i)23-s + (−12.5 + 21.6i)25-s + (64.5 − 111. i)29-s + (93.5 + 161. i)31-s − 110·35-s − 34·37-s + (132 + 228. i)41-s + (−221.5 + 383. i)43-s + (304.5 − 527. i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.593 − 1.02i)7-s + (−0.123 + 0.213i)11-s + (−0.181 − 0.314i)13-s + 1.07·17-s − 0.0482·19-s + (0.829 + 1.43i)23-s + (−0.100 + 0.173i)25-s + (0.413 − 0.715i)29-s + (0.541 + 0.938i)31-s − 0.531·35-s − 0.151·37-s + (0.502 + 0.870i)41-s + (−0.785 + 1.36i)43-s + (0.945 − 1.63i)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: $\chi_{1620} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.357105999\)
\(L(\frac12)\) \(\approx\) \(2.357105999\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11 + 19.0i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (4.5 - 7.79i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (8.5 + 14.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 75T + 4.91e3T^{2} \)
19 \( 1 + 4T + 6.85e3T^{2} \)
23 \( 1 + (-91.5 - 158. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-64.5 + 111. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-93.5 - 161. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 34T + 5.06e4T^{2} \)
41 \( 1 + (-132 - 228. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (221.5 - 383. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-304.5 + 527. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 228T + 1.48e5T^{2} \)
59 \( 1 + (-30 - 51.9i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-227 + 393. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-122 - 211. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 444T + 3.57e5T^{2} \)
73 \( 1 - 398T + 3.89e5T^{2} \)
79 \( 1 + (-174.5 + 302. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-519 + 898. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 852T + 7.04e5T^{2} \)
97 \( 1 + (457 - 791. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−8.887021347989308177739519677577, −7.935457598451804144186668615434, −7.56378067924706852336035093852, −6.66622860099694755241341941637, −5.46817162028381580440138851295, −4.83151475442300871000364450268, −3.91717419039542342930871540096, −3.00673134517374779776956232836, −1.51156191971170039694481270940, −0.73221605902118199363277345128, 0.824035797658204898708101768928, 2.20335936984373662627277247895, 2.93671301088842772093354486414, 4.10719458588245643611938530978, 5.09518968601489752763870683793, 5.78772693843080610431410256309, 6.73405443485970310721049845661, 7.56352253279900984347434398898, 8.434849524621675052822974007102, 8.927160289944172325203777224848

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