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Properties

Label 2-1620-45.29-c2-0-24
Degree $2$
Conductor $1620$
Sign $0.774 + 0.632i$
Analytic cond. $44.1418$
Root an. cond. $6.64392$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0689 + 4.99i)5-s + (−2.42 − 1.39i)7-s + (−15.7 − 9.07i)11-s + (−19.9 + 11.5i)13-s + 5.72·17-s + 23.1·19-s + (−0.135 − 0.235i)23-s + (−24.9 + 0.689i)25-s + (34.4 + 19.8i)29-s + (−23.6 − 41.0i)31-s + (6.82 − 12.2i)35-s − 34.8i·37-s + (−11.4 + 6.63i)41-s + (40.4 + 23.3i)43-s + (20.4 − 35.4i)47-s + ⋯
L(s)  = 1  + (0.0137 + 0.999i)5-s + (−0.345 − 0.199i)7-s + (−1.42 − 0.824i)11-s + (−1.53 + 0.885i)13-s + 0.336·17-s + 1.22·19-s + (−0.00590 − 0.0102i)23-s + (−0.999 + 0.0275i)25-s + (1.18 + 0.686i)29-s + (−0.764 − 1.32i)31-s + (0.194 − 0.348i)35-s − 0.942i·37-s + (−0.280 + 0.161i)41-s + (0.940 + 0.543i)43-s + (0.435 − 0.753i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.774 + 0.632i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.774 + 0.632i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.123544915\)
\(L(\frac12)\) \(\approx\) \(1.123544915\)
\(L(2)\) 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 \( 1 + (-0.0689 - 4.99i)T \)
good7 \( 1 + (2.42 + 1.39i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (15.7 + 9.07i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (19.9 - 11.5i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 5.72T + 289T^{2} \)
19 \( 1 - 23.1T + 361T^{2} \)
23 \( 1 + (0.135 + 0.235i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-34.4 - 19.8i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (23.6 + 41.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 34.8iT - 1.36e3T^{2} \)
41 \( 1 + (11.4 - 6.63i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-40.4 - 23.3i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-20.4 + 35.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 91.3T + 2.80e3T^{2} \)
59 \( 1 + (-68.2 + 39.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (15.5 - 27.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (5.98 - 3.45i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 81.5iT - 5.04e3T^{2} \)
73 \( 1 - 106. iT - 5.32e3T^{2} \)
79 \( 1 + (31.7 - 55.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-0.142 + 0.246i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 28.5iT - 7.92e3T^{2} \)
97 \( 1 + (-80.4 - 46.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.313203555780536779703770307415, −8.155525851878692542342458155421, −7.36195562528049880021659124310, −6.96323689522845114648180079857, −5.78740064710068651834555018339, −5.16099157860977036603523132365, −3.91499654442952511750358093010, −2.92294595704197794348286656476, −2.29274716801329512256576936074, −0.39530192224017506427680425247, 0.823548246831678816500947861841, 2.31053761442247988724982982140, 3.11635675199010142747503545683, 4.58292082785393288457466505832, 5.15888437013209632181594217505, 5.72631882580709587392262872285, 7.19484110857766240722300500399, 7.65714476501490410224668815877, 8.444290453356489553009949904201, 9.413938455995443925958033263855

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