Properties

Label 2-1620-15.14-c2-0-4
Degree $2$
Conductor $1620$
Sign $-0.998 + 0.0622i$
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  + (−4.99 + 0.311i)5-s + 4.65i·7-s + 12.0i·11-s + 8.46i·13-s + 1.25·17-s + 16.4·19-s − 2.38·23-s + (24.8 − 3.10i)25-s + 23.9i·29-s − 5.25·31-s + (−1.44 − 23.2i)35-s − 25.1i·37-s − 29.8i·41-s + 25.5i·43-s − 64.9·47-s + ⋯
L(s)  = 1  + (−0.998 + 0.0622i)5-s + 0.665i·7-s + 1.09i·11-s + 0.651i·13-s + 0.0736·17-s + 0.867·19-s − 0.103·23-s + (0.992 − 0.124i)25-s + 0.827i·29-s − 0.169·31-s + (−0.0413 − 0.663i)35-s − 0.680i·37-s − 0.727i·41-s + 0.593i·43-s − 1.38·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0622i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.998 + 0.0622i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.998 + 0.0622i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5729721150\)
\(L(\frac12)\) \(\approx\) \(0.5729721150\)
\(L(2)\) 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 \( 1 + (4.99 - 0.311i)T \)
good7 \( 1 - 4.65iT - 49T^{2} \)
11 \( 1 - 12.0iT - 121T^{2} \)
13 \( 1 - 8.46iT - 169T^{2} \)
17 \( 1 - 1.25T + 289T^{2} \)
19 \( 1 - 16.4T + 361T^{2} \)
23 \( 1 + 2.38T + 529T^{2} \)
29 \( 1 - 23.9iT - 841T^{2} \)
31 \( 1 + 5.25T + 961T^{2} \)
37 \( 1 + 25.1iT - 1.36e3T^{2} \)
41 \( 1 + 29.8iT - 1.68e3T^{2} \)
43 \( 1 - 25.5iT - 1.84e3T^{2} \)
47 \( 1 + 64.9T + 2.20e3T^{2} \)
53 \( 1 + 71.4T + 2.80e3T^{2} \)
59 \( 1 - 18.5iT - 3.48e3T^{2} \)
61 \( 1 - 19.1T + 3.72e3T^{2} \)
67 \( 1 - 81.0iT - 4.48e3T^{2} \)
71 \( 1 + 21.9iT - 5.04e3T^{2} \)
73 \( 1 + 109. iT - 5.32e3T^{2} \)
79 \( 1 + 134.T + 6.24e3T^{2} \)
83 \( 1 + 15.6T + 6.88e3T^{2} \)
89 \( 1 - 79.4iT - 7.92e3T^{2} \)
97 \( 1 - 130. iT - 9.40e3T^{2} \)
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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

−9.457648666702704531460573949178, −8.887876554683059831934616806398, −7.933638855290497519370839474464, −7.28054450202785280368634768557, −6.57160650369970258233695756675, −5.39000653904169495261737167750, −4.62556325960652652830401336206, −3.73783506083536981246671178300, −2.71332963337563208726267429661, −1.51259954090591246968757114511, 0.17496730127465045914190993389, 1.15676995086725380187558174880, 2.96844527063485614681914477756, 3.59012975493983806675161667308, 4.53632179592387648811513616132, 5.47556935769975859852984638174, 6.44195440093976785250233899758, 7.36865917979190112490602160270, 8.017752831025993281543321643142, 8.566909663175379025791524365532

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