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Properties

Label 2-3040-8.5-c1-0-15
Degree $2$
Conductor $3040$
Sign $-0.805 - 0.593i$
Analytic cond. $24.2745$
Root an. cond. $4.92691$
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  + 1.53i·3-s + i·5-s + 0.194·7-s + 0.650·9-s − 5.33i·11-s + 0.412i·13-s − 1.53·15-s − 1.18·17-s + i·19-s + 0.298i·21-s − 7.83·23-s − 25-s + 5.59i·27-s + 2.58i·29-s − 0.174·31-s + ⋯
L(s)  = 1  + 0.884i·3-s + 0.447i·5-s + 0.0736·7-s + 0.216·9-s − 1.60i·11-s + 0.114i·13-s − 0.395·15-s − 0.288·17-s + 0.229i·19-s + 0.0651i·21-s − 1.63·23-s − 0.200·25-s + 1.07i·27-s + 0.480i·29-s − 0.0312·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $-0.805 - 0.593i$
Analytic conductor: \(24.2745\)
Root analytic conductor: \(4.92691\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3040} (1521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :1/2),\ -0.805 - 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.256491090\)
\(L(\frac12)\) \(\approx\) \(1.256491090\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - iT \)
19 \( 1 - iT \)
good3 \( 1 - 1.53iT - 3T^{2} \)
7 \( 1 - 0.194T + 7T^{2} \)
11 \( 1 + 5.33iT - 11T^{2} \)
13 \( 1 - 0.412iT - 13T^{2} \)
17 \( 1 + 1.18T + 17T^{2} \)
23 \( 1 + 7.83T + 23T^{2} \)
29 \( 1 - 2.58iT - 29T^{2} \)
31 \( 1 + 0.174T + 31T^{2} \)
37 \( 1 - 11.1iT - 37T^{2} \)
41 \( 1 - 5.96T + 41T^{2} \)
43 \( 1 - 11.2iT - 43T^{2} \)
47 \( 1 + 6.77T + 47T^{2} \)
53 \( 1 + 4.93iT - 53T^{2} \)
59 \( 1 - 12.0iT - 59T^{2} \)
61 \( 1 - 11.5iT - 61T^{2} \)
67 \( 1 - 7.53iT - 67T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 - 0.911T + 73T^{2} \)
79 \( 1 - 0.990T + 79T^{2} \)
83 \( 1 + 2.93iT - 83T^{2} \)
89 \( 1 - 6.49T + 89T^{2} \)
97 \( 1 + 4.64T + 97T^{2} \)
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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

−9.031169519297865566286723064511, −8.322088946865211816313472827187, −7.68093088601441680820622948092, −6.54407563942647270518446947296, −6.04377987620735315607459384729, −5.10938114535882186757817904641, −4.23693857737482708234077517223, −3.53556754415382859091455430441, −2.75114375632896294440094961696, −1.33585126253459244880906368411, 0.39073379336118009350913321690, 1.87787739457846561079725063513, 2.14815074293538066614566279529, 3.80980676580924258336992635249, 4.49508162412747018161146348149, 5.34547394369099411086336663919, 6.34753783490000215585902110469, 6.93995577807788212993321259295, 7.76802621701221037567140933812, 8.070538402612714436704554756947

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