Properties

Label 2-2520-2520.293-c0-0-6
Degree $2$
Conductor $2520$
Sign $-0.999 + 0.0136i$
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  + (0.258 − 0.965i)2-s + (−0.923 − 0.382i)3-s + (−0.866 − 0.499i)4-s + (−0.130 − 0.991i)5-s + (−0.608 + 0.793i)6-s + (0.965 + 0.258i)7-s + (−0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.991 − 0.130i)10-s + (0.608 + 0.793i)12-s + (0.739 − 0.198i)13-s + (0.499 − 0.866i)14-s + (−0.258 + 0.965i)15-s + (0.500 + 0.866i)16-s + (0.866 − 0.5i)18-s − 1.58i·19-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.923 − 0.382i)3-s + (−0.866 − 0.499i)4-s + (−0.130 − 0.991i)5-s + (−0.608 + 0.793i)6-s + (0.965 + 0.258i)7-s + (−0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.991 − 0.130i)10-s + (0.608 + 0.793i)12-s + (0.739 − 0.198i)13-s + (0.499 − 0.866i)14-s + (−0.258 + 0.965i)15-s + (0.500 + 0.866i)16-s + (0.866 − 0.5i)18-s − 1.58i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.999 + 0.0136i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ -0.999 + 0.0136i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9111852445\)
\(L(\frac12)\) \(\approx\) \(0.9111852445\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (0.923 + 0.382i)T \)
5 \( 1 + (0.130 + 0.991i)T \)
7 \( 1 + (-0.965 - 0.258i)T \)
good11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.739 + 0.198i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + 1.58iT - T^{2} \)
23 \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.608 - 1.05i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + 0.517iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (1.78 + 0.478i)T + (0.866 + 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.866 - 0.5i)T^{2} \)
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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

−8.612576401936578303860744035093, −8.328920962990001651833048288930, −7.16777292972667881485513156026, −6.03515197745377091295946355329, −5.44106034964736829867923613706, −4.53150227653003295164831969020, −4.34206814037494873074555758821, −2.65754444491699532667290168595, −1.63799016665483257763204014756, −0.68852598260776806348650820073, 1.57966703951971425948338034377, 3.54394611578825399116289825106, 3.90295771563784624215412871659, 4.94514729471762721283664766265, 5.76723517429371060101736998203, 6.21951948348558292809005775750, 7.15878008365798890262589761662, 7.72402640832334114445339868459, 8.436117578145877596114860545193, 9.605471423636193657510239113654

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