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Properties

Label 2-1620-9.5-c2-0-25
Degree $2$
Conductor $1620$
Sign $0.173 + 0.984i$
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  + (1.93 + 1.11i)5-s + (−5.85 − 10.1i)7-s + (8.40 − 4.85i)11-s + (0.145 − 0.252i)13-s + 15i·17-s + 12.4·19-s + (29.8 + 17.2i)23-s + (2.5 + 4.33i)25-s + (3.21 − 1.85i)29-s + (−3.20 + 5.55i)31-s − 26.1i·35-s + 24.2·37-s + (−21.2 − 12.2i)41-s + (−37.9 − 65.7i)43-s + (22.0 − 12.7i)47-s + ⋯
L(s)  = 1  + (0.387 + 0.223i)5-s + (−0.836 − 1.44i)7-s + (0.764 − 0.441i)11-s + (0.0112 − 0.0194i)13-s + 0.882i·17-s + 0.653·19-s + (1.29 + 0.748i)23-s + (0.100 + 0.173i)25-s + (0.110 − 0.0639i)29-s + (−0.103 + 0.179i)31-s − 0.748i·35-s + 0.655·37-s + (−0.518 − 0.299i)41-s + (−0.883 − 1.52i)43-s + (0.468 − 0.270i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.173 + 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.853328134\)
\(L(\frac12)\) \(\approx\) \(1.853328134\)
\(L(2)\) 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 \( 1 + (-1.93 - 1.11i)T \)
good7 \( 1 + (5.85 + 10.1i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-8.40 + 4.85i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-0.145 + 0.252i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 15iT - 289T^{2} \)
19 \( 1 - 12.4T + 361T^{2} \)
23 \( 1 + (-29.8 - 17.2i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-3.21 + 1.85i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (3.20 - 5.55i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 24.2T + 1.36e3T^{2} \)
41 \( 1 + (21.2 + 12.2i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (37.9 + 65.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-22.0 + 12.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 64.0iT - 2.80e3T^{2} \)
59 \( 1 + (61.5 + 35.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (39.7 + 68.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 91.9iT - 5.04e3T^{2} \)
73 \( 1 - 13.1T + 5.32e3T^{2} \)
79 \( 1 + (-73.7 - 127. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-121. + 70.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 105. iT - 7.92e3T^{2} \)
97 \( 1 + (7.70 + 13.3i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.234288249162676730638971067366, −8.216439746673671661266507431227, −7.20512605213581839719193275490, −6.73640472284665042138157221844, −5.93042234764239060231920540653, −4.85481825069202634154697306571, −3.65883472159432653571517677727, −3.31224737598937148803839875727, −1.64868193313332572271664339107, −0.56414818573800756812641238608, 1.14639922806613731160237692266, 2.50829603504071682419794884125, 3.13072826809635451213120786783, 4.54480362799715087736749518117, 5.32156224587966397227062314296, 6.20086146049356826847608065558, 6.77643568301557852539368230464, 7.82503978503371252751925753312, 9.026582945171365489417672420365, 9.179912789376513905536965888228

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