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Properties

Label 2-1620-9.7-c3-0-19
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 + (11.6 − 20.1i)7-s + (−16.8 + 29.2i)11-s + (44.4 + 76.9i)13-s + 108.·17-s + 21.4·19-s + (−42.9 − 74.4i)23-s + (−12.5 + 21.6i)25-s + (9.48 − 16.4i)29-s + (83.1 + 144. i)31-s − 116.·35-s − 167.·37-s + (−189. − 327. i)41-s + (−190. + 329. i)43-s + (−123. + 213. i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.629 − 1.09i)7-s + (−0.462 + 0.801i)11-s + (0.948 + 1.64i)13-s + 1.55·17-s + 0.259·19-s + (−0.389 − 0.674i)23-s + (−0.100 + 0.173i)25-s + (0.0607 − 0.105i)29-s + (0.481 + 0.834i)31-s − 0.563·35-s − 0.744·37-s + (−0.720 − 1.24i)41-s + (−0.674 + 1.16i)43-s + (−0.383 + 0.663i)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: $\chi_{1620} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.064361788\)
\(L(\frac12)\) \(\approx\) \(2.064361788\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.6 + 20.1i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (16.8 - 29.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-44.4 - 76.9i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 108.T + 4.91e3T^{2} \)
19 \( 1 - 21.4T + 6.85e3T^{2} \)
23 \( 1 + (42.9 + 74.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-9.48 + 16.4i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-83.1 - 144. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 167.T + 5.06e4T^{2} \)
41 \( 1 + (189. + 327. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (190. - 329. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (123. - 213. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 284.T + 1.48e5T^{2} \)
59 \( 1 + (-363. - 629. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (235. - 407. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-375. - 651. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 864.T + 3.57e5T^{2} \)
73 \( 1 + 1.00e3T + 3.89e5T^{2} \)
79 \( 1 + (-419. + 726. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (238. - 412. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 554.T + 7.04e5T^{2} \)
97 \( 1 + (-523. + 907. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−9.036569755557379501589756947617, −8.327643201212624807083715047532, −7.49715094882167476310114692559, −6.94581039359167802295350414513, −5.88206056792697205774634346509, −4.75249353145908958988967366951, −4.28220512924816817643782390995, −3.30801704424896867491149821516, −1.75623889578110364199769461668, −1.06384623848320575784381009957, 0.50592167780101154876316278881, 1.74848525818236624577689892458, 3.11985737178090491985655281567, 3.43062724917144963766635850832, 5.11225162745278794668809177213, 5.58713713945179964305886041537, 6.26600665312262676260034699081, 7.64949947124893756919174048877, 8.132968094253878730324305270590, 8.636734693031100840481517506481

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