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Properties

Label 2-2520-2520.2029-c0-0-7
Degree $2$
Conductor $2520$
Sign $-0.906 + 0.422i$
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  + (0.866 − 0.5i)2-s + (0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (−0.707 − 0.707i)5-s + (0.258 − 0.965i)6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s − 1.00i·9-s + (−0.965 − 0.258i)10-s + (−0.258 − 0.965i)12-s + (−1.22 − 0.707i)13-s + (−0.499 + 0.866i)14-s − 1.00·15-s + (−0.5 − 0.866i)16-s + (−0.500 − 0.866i)18-s + 0.517·19-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (−0.707 − 0.707i)5-s + (0.258 − 0.965i)6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s − 1.00i·9-s + (−0.965 − 0.258i)10-s + (−0.258 − 0.965i)12-s + (−1.22 − 0.707i)13-s + (−0.499 + 0.866i)14-s − 1.00·15-s + (−0.5 − 0.866i)16-s + (−0.500 − 0.866i)18-s + 0.517·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.906 + 0.422i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (2029, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ -0.906 + 0.422i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.841095004\)
\(L(\frac12)\) \(\approx\) \(1.841095004\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
good11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 0.517T + T^{2} \)
23 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - 1.73T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−8.948745407946266898792103355178, −7.81229737795408855359043248703, −7.26035960844685839390702844381, −6.47344478007787475006545129792, −5.43631966549747096674955100258, −4.85144894246307206655767678446, −3.56708211409875835378910969579, −3.13743103027962582729479615872, −2.16520029951321717550769373687, −0.820003469803503444506877484270, 2.50156684372160513870310925946, 3.02364164082644446447693374005, 3.88415936102061332209783359621, 4.49593135173543455701083598967, 5.31308738602384513798855200464, 6.58297695429233348371553541810, 7.10478915032948939157870407181, 7.63345502476727761524521801436, 8.562370949862687615197306255382, 9.362195040193264947020182501427

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