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Properties

Label 2-2520-2520.293-c0-0-4
Degree $2$
Conductor $2520$
Sign $0.872 + 0.488i$
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 + (−0.382 − 0.923i)3-s + (−0.866 − 0.499i)4-s + (−0.923 − 0.382i)5-s + (0.991 − 0.130i)6-s + (0.965 + 0.258i)7-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.608 − 0.793i)10-s + (−0.130 + 0.991i)12-s + (0.739 − 0.198i)13-s + (−0.499 + 0.866i)14-s + i·15-s + (0.500 + 0.866i)16-s + (−0.500 − 0.866i)18-s − 0.261i·19-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.382 − 0.923i)3-s + (−0.866 − 0.499i)4-s + (−0.923 − 0.382i)5-s + (0.991 − 0.130i)6-s + (0.965 + 0.258i)7-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.608 − 0.793i)10-s + (−0.130 + 0.991i)12-s + (0.739 − 0.198i)13-s + (−0.499 + 0.866i)14-s + i·15-s + (0.500 + 0.866i)16-s + (−0.500 − 0.866i)18-s − 0.261i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.872 + 0.488i$
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: \((2,\ 2520,\ (\ :0),\ 0.872 + 0.488i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7464693269\)
\(L(\frac12)\) \(\approx\) \(0.7464693269\)
\(L(1)\) 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 \( 1 + (0.382 + 0.923i)T \)
5 \( 1 + (0.923 + 0.382i)T \)
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 \( 1 + 0.261iT - T^{2} \)
23 \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (0.991 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + 1.93iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
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

−8.618159140106370480989300984313, −8.069939237399828439058833406548, −7.70109325307319113264292624698, −6.78631457216355814785258336465, −6.11177876375817732162205212423, −5.15364067456802435951175705017, −4.68430718770571630818271622403, −3.52156619576612235286721814370, −1.86820169698814064000117125532, −0.72321670365527993664146566050, 1.13651310756066894116329087623, 2.62222006087666582939404022932, 3.70586448304964562468035629544, 4.09913901427167676237283628604, 4.90722960712440956022159039944, 5.78647089524455355761587266206, 7.05174860975355057576084051741, 7.936788071424043272453523910206, 8.538526960822909879729880589425, 9.197218724288004002622526209993

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