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Properties

Label 2-3040-8.5-c1-0-66
Degree $2$
Conductor $3040$
Sign $-0.920 - 0.391i$
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  − 2.02i·3-s − i·5-s − 1.28·7-s − 1.09·9-s − 2.17i·11-s − 0.732i·13-s − 2.02·15-s − 0.143·17-s − i·19-s + 2.59i·21-s + 0.625·23-s − 25-s − 3.85i·27-s − 6.12i·29-s + 2.33·31-s + ⋯
L(s)  = 1  − 1.16i·3-s − 0.447i·5-s − 0.485·7-s − 0.365·9-s − 0.654i·11-s − 0.203i·13-s − 0.522·15-s − 0.0346·17-s − 0.229i·19-s + 0.567i·21-s + 0.130·23-s − 0.200·25-s − 0.741i·27-s − 1.13i·29-s + 0.419·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $-0.920 - 0.391i$
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: \((2,\ 3040,\ (\ :1/2),\ -0.920 - 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.019336781\)
\(L(\frac12)\) \(\approx\) \(1.019336781\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.02iT - 3T^{2} \)
7 \( 1 + 1.28T + 7T^{2} \)
11 \( 1 + 2.17iT - 11T^{2} \)
13 \( 1 + 0.732iT - 13T^{2} \)
17 \( 1 + 0.143T + 17T^{2} \)
23 \( 1 - 0.625T + 23T^{2} \)
29 \( 1 + 6.12iT - 29T^{2} \)
31 \( 1 - 2.33T + 31T^{2} \)
37 \( 1 + 2.46iT - 37T^{2} \)
41 \( 1 + 1.52T + 41T^{2} \)
43 \( 1 - 5.44iT - 43T^{2} \)
47 \( 1 + 9.83T + 47T^{2} \)
53 \( 1 + 11.2iT - 53T^{2} \)
59 \( 1 - 0.188iT - 59T^{2} \)
61 \( 1 - 6.59iT - 61T^{2} \)
67 \( 1 - 1.58iT - 67T^{2} \)
71 \( 1 + 8.09T + 71T^{2} \)
73 \( 1 + 4.18T + 73T^{2} \)
79 \( 1 + 1.57T + 79T^{2} \)
83 \( 1 - 5.97iT - 83T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 - 3.88T + 97T^{2} \)
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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

−8.119121046138345440059157968809, −7.66162574868059335006549846167, −6.65571570275533596831784017372, −6.26960496261216413416539794915, −5.37089002832250943542521270736, −4.39504784496627841973227152815, −3.33953727131298764035455517272, −2.39117526442340074099292410340, −1.32913775337826743045594455888, −0.32314334289394338530212714595, 1.66349097307542143509662647400, 2.96570117550762875390159895089, 3.59672072513869448343005309060, 4.50835641992061331058625760573, 5.08090399860453310638972264380, 6.11746981895916200484284245363, 6.86639636823378288147137383892, 7.56845511784018525304514473734, 8.612593386575021546027815419515, 9.296386137569350914504835833500

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