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Properties

Label 2-3040-76.75-c1-0-73
Degree $2$
Conductor $3040$
Sign $0.526 + 0.850i$
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  + 2.82·3-s + 5-s − 2i·7-s + 5.00·9-s − 2i·11-s − 2.82i·13-s + 2.82·15-s + 2·17-s + (−4.24 − i)19-s − 5.65i·21-s − 2i·23-s + 25-s + 5.65·27-s − 2.82i·29-s − 5.65i·33-s + ⋯
L(s)  = 1  + 1.63·3-s + 0.447·5-s − 0.755i·7-s + 1.66·9-s − 0.603i·11-s − 0.784i·13-s + 0.730·15-s + 0.485·17-s + (−0.973 − 0.229i)19-s − 1.23i·21-s − 0.417i·23-s + 0.200·25-s + 1.08·27-s − 0.525i·29-s − 0.984i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $0.526 + 0.850i$
Analytic conductor: \(24.2745\)
Root analytic conductor: \(4.92691\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3040} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :1/2),\ 0.526 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.613809190\)
\(L(\frac12)\) \(\approx\) \(3.613809190\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
19 \( 1 + (4.24 + i)T \)
good3 \( 1 - 2.82T + 3T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + 2.82iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
23 \( 1 + 2iT - 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 8.48iT - 37T^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 - 2.82iT - 53T^{2} \)
59 \( 1 + 2.82T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.447876937867930639440048272024, −8.053993851492553035385324127995, −7.32306770450329408583350417065, −6.47794445077271436502451548980, −5.55608324974801100426673808615, −4.40728894724789740006766033307, −3.70325698488763614204081985149, −2.89501058052352684384887723667, −2.14728188123133779025634767400, −0.897152694373119918809416983035, 1.68195681931192430042981520448, 2.20651942645308266424485423111, 3.10156124820958474100796261771, 3.95355791295746228520322866427, 4.83047347321176781606164322749, 5.82797861129935461662775168834, 6.79202787890157331585895451237, 7.42083699820940409677266233918, 8.407101059050924290717144148571, 8.699470680382877113016817073541

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