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Properties

Label 2-3040-760.339-c0-0-1
Degree $2$
Conductor $3040$
Sign $0.513 + 0.858i$
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  + (0.173 − 0.984i)5-s + (0.173 + 0.300i)7-s + (0.766 − 0.642i)9-s + (0.766 − 1.32i)11-s + (0.939 + 0.342i)13-s + (−0.766 − 0.642i)19-s + (0.326 + 1.85i)23-s + (−0.939 − 0.342i)25-s + (0.326 − 0.118i)35-s − 1.87·37-s + (−1.43 + 0.524i)41-s + (−0.5 − 0.866i)45-s + (0.766 − 0.642i)47-s + (0.439 − 0.761i)49-s + (0.0603 + 0.342i)53-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)5-s + (0.173 + 0.300i)7-s + (0.766 − 0.642i)9-s + (0.766 − 1.32i)11-s + (0.939 + 0.342i)13-s + (−0.766 − 0.642i)19-s + (0.326 + 1.85i)23-s + (−0.939 − 0.342i)25-s + (0.326 − 0.118i)35-s − 1.87·37-s + (−1.43 + 0.524i)41-s + (−0.5 − 0.866i)45-s + (0.766 − 0.642i)47-s + (0.439 − 0.761i)49-s + (0.0603 + 0.342i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $0.513 + 0.858i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3040} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :0),\ 0.513 + 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.466922956\)
\(L(\frac12)\) \(\approx\) \(1.466922956\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
good3 \( 1 + (-0.766 + 0.642i)T^{2} \)
7 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.87T + T^{2} \)
41 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
53 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-0.173 - 0.984i)T^{2} \)
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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

−8.831204430282686151048658629073, −8.356167425987301212531909120440, −7.18422841045137071596650748255, −6.45647445316805296303329126534, −5.71473363315743450351330412483, −4.99811793399565669109892879695, −3.89127048210134237476464856685, −3.48350150715923967869680499527, −1.81483610072847258740953384780, −1.03188143779678336927879676959, 1.58810479876707496338393692004, 2.30955032332008877390669983433, 3.59697862205992056583519258690, 4.25039907990385125908785011267, 5.08545707655593841234401234439, 6.26235021484750890794680281067, 6.79827287123337314864700455823, 7.35525529860549122109266265075, 8.230914597813344408081200321619, 8.979677881980374828421181767864

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