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Properties

Label 2-1020-5.4-c1-0-9
Degree $2$
Conductor $1020$
Sign $0.829 + 0.559i$
Analytic cond. $8.14474$
Root an. cond. $2.85389$
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  − i·3-s + (1.85 + 1.25i)5-s + 1.12i·7-s − 9-s − 0.781·11-s − 6.90i·13-s + (1.25 − 1.85i)15-s − i·17-s + 4.99·19-s + 1.12·21-s − 2.69i·23-s + (1.87 + 4.63i)25-s + i·27-s + 4.48·29-s + 5.36·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.829 + 0.559i)5-s + 0.424i·7-s − 0.333·9-s − 0.235·11-s − 1.91i·13-s + (0.322 − 0.478i)15-s − 0.242i·17-s + 1.14·19-s + 0.245·21-s − 0.562i·23-s + (0.374 + 0.927i)25-s + 0.192i·27-s + 0.833·29-s + 0.963·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.829 + 0.559i$
Analytic conductor: \(8.14474\)
Root analytic conductor: \(2.85389\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1020} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1020,\ (\ :1/2),\ 0.829 + 0.559i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.867593728\)
\(L(\frac12)\) \(\approx\) \(1.867593728\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (-1.85 - 1.25i)T \)
17 \( 1 + iT \)
good7 \( 1 - 1.12iT - 7T^{2} \)
11 \( 1 + 0.781T + 11T^{2} \)
13 \( 1 + 6.90iT - 13T^{2} \)
19 \( 1 - 4.99T + 19T^{2} \)
23 \( 1 + 2.69iT - 23T^{2} \)
29 \( 1 - 4.48T + 29T^{2} \)
31 \( 1 - 5.36T + 31T^{2} \)
37 \( 1 + 1.87iT - 37T^{2} \)
41 \( 1 - 12.1T + 41T^{2} \)
43 \( 1 - 4.10iT - 43T^{2} \)
47 \( 1 - 12.1iT - 47T^{2} \)
53 \( 1 + 7.12iT - 53T^{2} \)
59 \( 1 + 8.61T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 + 8.61iT - 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + 7.29iT - 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 - 13.0iT - 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + 18.2iT - 97T^{2} \)
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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

−9.915801658801476165121310326010, −9.112038792161597229563888113327, −8.001234195281814611135356001366, −7.49131762258617275072314181684, −6.29833757025857455608529096208, −5.77831784910159371633681479900, −4.87982971422130821486258705147, −3.05762742661400043204201483278, −2.61442909258455375203497984183, −1.01770644586158768833246735148, 1.30445875257028939355857812239, 2.59519455053895040382739056239, 4.01070555226233830347381057344, 4.71460746623800598662832313814, 5.66107777184493740711690035982, 6.55052543829634552243760601658, 7.50397179181763375031125998044, 8.667544914260807429953378420835, 9.281867785265370326489583635159, 9.907782797180915955844132660634

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