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Properties

Label 2-1620-9.7-c3-0-28
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 + (−8.5 + 14.7i)7-s + (15 − 25.9i)11-s + (30.5 + 52.8i)13-s − 120·17-s − 43·19-s + (−45 − 77.9i)23-s + (−12.5 + 21.6i)25-s + (45 − 77.9i)29-s + (−4 − 6.92i)31-s + 85·35-s + 317·37-s + (15 + 25.9i)41-s + (110 − 190. i)43-s + (90 − 155. i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.458 + 0.794i)7-s + (0.411 − 0.712i)11-s + (0.650 + 1.12i)13-s − 1.71·17-s − 0.519·19-s + (−0.407 − 0.706i)23-s + (−0.100 + 0.173i)25-s + (0.288 − 0.499i)29-s + (−0.0231 − 0.0401i)31-s + 0.410·35-s + 1.40·37-s + (0.0571 + 0.0989i)41-s + (0.390 − 0.675i)43-s + (0.279 − 0.483i)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

\(L(2)\) \(\approx\) \(1.428014750\)
\(L(\frac12)\) \(\approx\) \(1.428014750\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.5 - 14.7i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-15 + 25.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-30.5 - 52.8i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 120T + 4.91e3T^{2} \)
19 \( 1 + 43T + 6.85e3T^{2} \)
23 \( 1 + (45 + 77.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-45 + 77.9i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 317T + 5.06e4T^{2} \)
41 \( 1 + (-15 - 25.9i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-110 + 190. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-90 + 155. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 630T + 1.48e5T^{2} \)
59 \( 1 + (-420 - 727. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (299.5 - 518. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (53.5 + 92.6i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 210T + 3.57e5T^{2} \)
73 \( 1 + 421T + 3.89e5T^{2} \)
79 \( 1 + (176.5 - 305. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-675 + 1.16e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + (-498.5 + 863. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−8.758564152916730361485598514849, −8.583972458265926972867258102255, −7.28079275959259560500989343637, −6.21004499864522116792549263053, −6.07135376585900751757701606263, −4.54593645428264915969114211424, −4.08329788951541362163264855892, −2.78547446229487855110917349061, −1.85316408164599498483002928960, −0.44042020706527729357166089941, 0.72535224945872648647866593279, 2.05761745624867851164261772657, 3.21914617192825412487759132227, 4.03619744595898497975071600206, 4.81819329265286837976683784383, 6.16144763114791683265952609151, 6.63185884672655922927580328079, 7.52676892481923726933875965066, 8.203880469738023489568000674153, 9.226179083429175188659459723174

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