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Properties

Label 2-1620-9.4-c3-0-27
Degree $2$
Conductor $1620$
Sign $0.642 + 0.766i$
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 + (−0.292 − 0.506i)7-s + (5.96 + 10.3i)11-s + (−26.8 + 46.5i)13-s + 52.2·17-s − 144.·19-s + (27.4 − 47.5i)23-s + (−12.5 − 21.6i)25-s + (−100. − 174. i)29-s + (108. − 188. i)31-s − 2.92·35-s + 318.·37-s + (−187. + 325. i)41-s + (19.0 + 33.0i)43-s + (228. + 396. i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.0157 − 0.0273i)7-s + (0.163 + 0.282i)11-s + (−0.573 + 0.993i)13-s + 0.745·17-s − 1.74·19-s + (0.248 − 0.430i)23-s + (−0.100 − 0.173i)25-s + (−0.644 − 1.11i)29-s + (0.629 − 1.09i)31-s − 0.0141·35-s + 1.41·37-s + (−0.715 + 1.23i)41-s + (0.0677 + 0.117i)43-s + (0.709 + 1.22i)47-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(2)\) \(\approx\) \(1.837548231\)
\(L(\frac12)\) \(\approx\) \(1.837548231\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.292 + 0.506i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-5.96 - 10.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (26.8 - 46.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 52.2T + 4.91e3T^{2} \)
19 \( 1 + 144.T + 6.85e3T^{2} \)
23 \( 1 + (-27.4 + 47.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (100. + 174. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-108. + 188. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 318.T + 5.06e4T^{2} \)
41 \( 1 + (187. - 325. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-19.0 - 33.0i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-228. - 396. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 356.T + 1.48e5T^{2} \)
59 \( 1 + (-270. + 468. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-205. - 355. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (181. - 313. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 175.T + 3.57e5T^{2} \)
73 \( 1 + 105.T + 3.89e5T^{2} \)
79 \( 1 + (178. + 309. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (590. + 1.02e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 64.1T + 7.04e5T^{2} \)
97 \( 1 + (545. + 944. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−8.940181134265904733412355543341, −8.180943806069397140465138396889, −7.34559586399428799484641489361, −6.41569793200712027001742859853, −5.78182594590019247285891382399, −4.50314520713484535891597542693, −4.19028629309837070260322413928, −2.64709323086019259878169747571, −1.82809311099149549059113904197, −0.50278370394238192949693730733, 0.836164052138600444876357345632, 2.16181813028282756094442570434, 3.09220167152638461802601208737, 4.00970247615429339032918132948, 5.18638856002034382191209974246, 5.82448166754789825262644926952, 6.79294895035806808559065634086, 7.47931299391926321445702933327, 8.417213129414684467947762178876, 9.046018022359926031809675530285

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