Properties

Label 2-2520-280.269-c0-0-2
Degree $2$
Conductor $2520$
Sign $-0.991 + 0.126i$
Analytic cond. $1.25764$
Root an. cond. $1.12144$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + 0.999·8-s − 0.999·10-s + (−1.5 − 0.866i)11-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (0.499 + 0.866i)20-s + 1.73i·22-s + (−0.499 − 0.866i)25-s + (−0.499 − 0.866i)28-s − 1.73i·29-s + (−1.5 − 0.866i)31-s + (−0.499 + 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + 0.999·8-s − 0.999·10-s + (−1.5 − 0.866i)11-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (0.499 + 0.866i)20-s + 1.73i·22-s + (−0.499 − 0.866i)25-s + (−0.499 − 0.866i)28-s − 1.73i·29-s + (−1.5 − 0.866i)31-s + (−0.499 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ -0.991 + 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4995994469\)
\(L(\frac12)\) \(\approx\) \(0.4995994469\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good11 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + 1.73iT - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−8.941085060753119121150840365324, −8.120698976982002192957729464377, −7.67685506422168550048407575987, −6.19188885543868243695005092675, −5.53744930580630495810310083325, −4.77909387233683719618008975357, −3.64967509166395519374445727408, −2.65723026404817183797940640211, −1.98719967937186098026503117062, −0.37090590848857770032086575914, 1.63141028656279175230985336625, 2.83320836993812420942846545150, 3.93035864884599878931306877547, 5.10914910299549229702469384201, 5.58577881417783905808875067267, 6.83640181730218129581832489682, 7.00142651918653986204386683242, 7.71422151452433866373053179238, 8.606402666192640700344768862547, 9.632920543496502484663595866130

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