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Properties

Label 2-3040-3040.1709-c0-0-2
Degree $2$
Conductor $3040$
Sign $0.773 + 0.634i$
Analytic cond. $1.51715$
Root an. cond. $1.23172$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.773 − 0.634i)2-s + (0.0750 − 0.181i)3-s + (0.195 − 0.980i)4-s + (−0.923 + 0.382i)5-s + (−0.0569 − 0.187i)6-s + (−0.471 − 0.881i)8-s + (0.679 + 0.679i)9-s + (−0.471 + 0.881i)10-s + (0.425 + 1.02i)11-s + (−0.162 − 0.108i)12-s + (0.536 + 0.222i)13-s + 0.196i·15-s + (−0.923 − 0.382i)16-s + (0.956 + 0.0942i)18-s + (0.923 + 0.382i)19-s + (0.195 + 0.980i)20-s + ⋯
L(s)  = 1  + (0.773 − 0.634i)2-s + (0.0750 − 0.181i)3-s + (0.195 − 0.980i)4-s + (−0.923 + 0.382i)5-s + (−0.0569 − 0.187i)6-s + (−0.471 − 0.881i)8-s + (0.679 + 0.679i)9-s + (−0.471 + 0.881i)10-s + (0.425 + 1.02i)11-s + (−0.162 − 0.108i)12-s + (0.536 + 0.222i)13-s + 0.196i·15-s + (−0.923 − 0.382i)16-s + (0.956 + 0.0942i)18-s + (0.923 + 0.382i)19-s + (0.195 + 0.980i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $0.773 + 0.634i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3040} (1709, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :0),\ 0.773 + 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.845993296\)
\(L(\frac12)\) \(\approx\) \(1.845993296\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.773 + 0.634i)T \)
5 \( 1 + (0.923 - 0.382i)T \)
19 \( 1 + (-0.923 - 0.382i)T \)
good3 \( 1 + (-0.0750 + 0.181i)T + (-0.707 - 0.707i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (-0.425 - 1.02i)T + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.536 - 0.222i)T + (0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (0.707 + 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.17 + 0.485i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.761 + 1.83i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.636 - 1.53i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 + (0.674 - 1.62i)T + (-0.707 - 0.707i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 - 1.91T + T^{2} \)
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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

−8.896127257752942838675040280903, −7.87030074479517008780602808164, −7.20246832666292095857423701645, −6.64948063028317014933802982216, −5.61519154424590565774996071348, −4.64156802202322436984657425789, −4.13420731376855595334571459664, −3.31777758352774097978827016384, −2.27121978707050885142336920998, −1.29713672214449542396498264868, 1.10350572633682419028023088950, 3.03071731232576440405789048946, 3.53835155394797628140168419133, 4.32578584826370271905905571975, 4.99876452004165507298958685874, 6.06489314151319316284276395283, 6.54584088578353128324416717769, 7.63734212463274084185445557611, 7.904210548537236384591311149227, 9.031342643543614067317985610241

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