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Properties

Label 2-2520-7.2-c1-0-26
Degree $2$
Conductor $2520$
Sign $0.740 + 0.671i$
Analytic cond. $20.1223$
Root an. cond. $4.48578$
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  + (0.5 − 0.866i)5-s + (0.647 + 2.56i)7-s + (−2.25 − 3.89i)11-s + 5.09·13-s + (1 + 1.73i)17-s + (1.54 − 2.67i)19-s + (2.89 − 5.01i)23-s + (−0.499 − 0.866i)25-s − 9.50·29-s + (−2.70 − 4.68i)31-s + (2.54 + 0.722i)35-s + (−3.54 + 6.14i)37-s + 6.59·41-s + 4.70·43-s + (5.04 − 8.74i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.244 + 0.969i)7-s + (−0.678 − 1.17i)11-s + 1.41·13-s + (0.242 + 0.420i)17-s + (0.354 − 0.614i)19-s + (0.604 − 1.04i)23-s + (−0.0999 − 0.173i)25-s − 1.76·29-s + (−0.485 − 0.841i)31-s + (0.430 + 0.122i)35-s + (−0.582 + 1.00i)37-s + 1.02·41-s + 0.717·43-s + (0.736 − 1.27i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.740 + 0.671i$
Analytic conductor: \(20.1223\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ 0.740 + 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.920125912\)
\(L(\frac12)\) \(\approx\) \(1.920125912\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.647 - 2.56i)T \)
good11 \( 1 + (2.25 + 3.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.09T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.54 + 2.67i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.89 + 5.01i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.50T + 29T^{2} \)
31 \( 1 + (2.70 + 4.68i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.54 - 6.14i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.59T + 41T^{2} \)
43 \( 1 - 4.70T + 43T^{2} \)
47 \( 1 + (-5.04 + 8.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.95 - 8.58i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.45 + 7.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0585 - 0.101i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (4.09 + 7.08i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.09 + 12.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 + (-2.04 + 3.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−8.856323573358329515205373084836, −8.259612551545470364433188872726, −7.43109206990443120552206714403, −6.17742138671314485306131673985, −5.77922041325265023123513404220, −5.08992922084969592325371878325, −3.92996529338040109809935485091, −3.02514490617598773867474268720, −2.02640998094659300254202942890, −0.73621567952473347233766027221, 1.15159312526146286875003159273, 2.15225372443919468151453960290, 3.48838066986046173692569630760, 4.01056668251374090412187968438, 5.20593220586131756362477123673, 5.77181259493710191116454274800, 6.97296404347239266697258971910, 7.37829380815196471995970904287, 8.032281497564281358916268700081, 9.177526349304718641567811070925

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