Properties

Label 2-1620-45.38-c1-0-9
Degree $2$
Conductor $1620$
Sign $0.979 + 0.203i$
Analytic cond. $12.9357$
Root an. cond. $3.59663$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.11 − 1.93i)5-s + (−0.366 − 1.36i)7-s + (3.87 + 2.23i)11-s + (−1.09 + 4.09i)13-s + (2.23 + 2.23i)17-s + 2i·19-s + (3.05 + 0.818i)23-s + (−2.5 + 4.33i)25-s + (2.23 − 3.87i)29-s + (−2 − 3.46i)31-s + (−2.23 + 2.23i)35-s + (−3 + 3i)37-s + (7.74 − 4.47i)41-s + (4.09 − 1.09i)43-s + (9.16 − 2.45i)47-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)5-s + (−0.138 − 0.516i)7-s + (1.16 + 0.674i)11-s + (−0.304 + 1.13i)13-s + (0.542 + 0.542i)17-s + 0.458i·19-s + (0.636 + 0.170i)23-s + (−0.5 + 0.866i)25-s + (0.415 − 0.719i)29-s + (−0.359 − 0.622i)31-s + (−0.377 + 0.377i)35-s + (−0.493 + 0.493i)37-s + (1.20 − 0.698i)41-s + (0.624 − 0.167i)43-s + (1.33 − 0.358i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.979 + 0.203i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ 0.979 + 0.203i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.619709473\)
\(L(\frac12)\) \(\approx\) \(1.619709473\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.11 + 1.93i)T \)
good7 \( 1 + (0.366 + 1.36i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-3.87 - 2.23i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.09 - 4.09i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-2.23 - 2.23i)T + 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (-3.05 - 0.818i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.23 + 3.87i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + (-7.74 + 4.47i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.09 + 1.09i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-9.16 + 2.45i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.23 + 2.23i)T - 53iT^{2} \)
59 \( 1 + (4.47 + 7.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.36 - 0.366i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.47iT - 71T^{2} \)
73 \( 1 + (-1 - i)T + 73iT^{2} \)
79 \( 1 + (5.19 + 3i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.45 - 9.16i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 + (-3.29 - 12.2i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.315444924558336240346998680617, −8.709215091331595154985143389026, −7.69480317545725018765320932831, −7.08464891491273576553616859591, −6.19524567092682532375950480894, −5.12412047277643606535670835699, −4.11009323155347410691756056155, −3.83596827603476500592920920576, −2.05057720287685639847181117873, −0.957419846163250766956597661213, 0.887878267050731601774804034136, 2.69549955708055606099580761112, 3.23232694926330784702014635706, 4.28859737023943266536068504640, 5.47494615781897302292602987886, 6.17992152112189714236923857926, 7.11668929436845628409572955632, 7.67024955202400337378370013308, 8.775264110121731177753698826664, 9.237995329652230967273924526799

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