Properties

Label 2-1620-45.2-c1-0-2
Degree $2$
Conductor $1620$
Sign $-0.548 - 0.835i$
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  + (2.19 − 0.448i)5-s + (−2.73 + 0.732i)7-s + (−2.44 − 1.41i)11-s + (−4.09 − 1.09i)13-s + (1.41 − 1.41i)17-s + 4i·19-s + (−2.07 + 7.72i)23-s + (4.59 − 1.96i)25-s + (−4.94 + 8.57i)29-s + (4 + 6.92i)31-s + (−5.65 + 2.82i)35-s + (−3 − 3i)37-s + (−1.22 + 0.707i)41-s + (3.10 + 11.5i)47-s + (0.866 − 0.5i)49-s + ⋯
L(s)  = 1  + (0.979 − 0.200i)5-s + (−1.03 + 0.276i)7-s + (−0.738 − 0.426i)11-s + (−1.13 − 0.304i)13-s + (0.342 − 0.342i)17-s + 0.917i·19-s + (−0.431 + 1.61i)23-s + (0.919 − 0.392i)25-s + (−0.919 + 1.59i)29-s + (0.718 + 1.24i)31-s + (−0.956 + 0.478i)35-s + (−0.493 − 0.493i)37-s + (−0.191 + 0.110i)41-s + (0.453 + 1.69i)47-s + (0.123 − 0.0714i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.548 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.548 - 0.835i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (917, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ -0.548 - 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7674196750\)
\(L(\frac12)\) \(\approx\) \(0.7674196750\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.19 + 0.448i)T \)
good7 \( 1 + (2.73 - 0.732i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (2.44 + 1.41i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.09 + 1.09i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (2.07 - 7.72i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (4.94 - 8.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 + 3i)T + 37iT^{2} \)
41 \( 1 + (1.22 - 0.707i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-3.10 - 11.5i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (7.07 + 7.07i)T + 53iT^{2} \)
59 \( 1 + (-1.41 - 2.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.92 + 10.9i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + (-7 + 7i)T - 73iT^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.5 - 3.10i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 1.41T + 89T^{2} \)
97 \( 1 + (4.09 - 1.09i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.641069189917754365995192449740, −9.143849459861745552140853277070, −8.056863689134916628409909085886, −7.25256383177007549114090242303, −6.33866525436698777507452135465, −5.49604736574711450119099808637, −5.07645756503856173118600350660, −3.45565965013656958940663654478, −2.78981070836599794742919372380, −1.55766333842511828498776983107, 0.27146543327488306504411358977, 2.21700420476205526053269984440, 2.73155364524024758348545088494, 4.11069311061086098812443509234, 5.03085479281297195818651391040, 5.97476652448043568130316028725, 6.67248047079219888415136567600, 7.36363874312639336758867702326, 8.363421677593722446432493373042, 9.401868717053423196267709664172

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