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Properties

Label 2-2520-2520.1973-c0-0-3
Degree $2$
Conductor $2520$
Sign $0.187 + 0.982i$
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  + (−0.965 + 0.258i)2-s + (−0.923 + 0.382i)3-s + (0.866 − 0.499i)4-s + (0.382 − 0.923i)5-s + (0.793 − 0.608i)6-s + (0.258 + 0.965i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (−0.130 + 0.991i)10-s + (−0.608 + 0.793i)12-s + (0.478 − 1.78i)13-s + (−0.499 − 0.866i)14-s + i·15-s + (0.500 − 0.866i)16-s + (−0.5 + 0.866i)18-s − 1.21i·19-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (−0.923 + 0.382i)3-s + (0.866 − 0.499i)4-s + (0.382 − 0.923i)5-s + (0.793 − 0.608i)6-s + (0.258 + 0.965i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (−0.130 + 0.991i)10-s + (−0.608 + 0.793i)12-s + (0.478 − 1.78i)13-s + (−0.499 − 0.866i)14-s + i·15-s + (0.500 − 0.866i)16-s + (−0.5 + 0.866i)18-s − 1.21i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.187 + 0.982i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ 0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4889453957\)
\(L(\frac12)\) \(\approx\) \(0.4889453957\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (0.923 - 0.382i)T \)
5 \( 1 + (-0.382 + 0.923i)T \)
7 \( 1 + (-0.258 - 0.965i)T \)
good11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.478 + 1.78i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + 1.21iT - T^{2} \)
23 \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.793 - 1.37i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.198 + 0.739i)T + (-0.866 + 0.5i)T^{2} \)
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

−8.902114546372265653018134631754, −8.339262793179812035273304098037, −7.60932216272706707372483449530, −6.39191841308055795304503098059, −5.80178774484001435414726487705, −5.36654873389507145171885688945, −4.46444267264550450583134779318, −2.94590097650049485732556246627, −1.75501866994657583404803790668, −0.50246318391336078818355363187, 1.51448997420254098275993909022, 2.02980200376602394579028886411, 3.62410381485552667657028106435, 4.28880441665573267633974103638, 5.84107284073668067332020350701, 6.36293749565419531644092073603, 7.02734062983082674153121837576, 7.62154477438433459025003489186, 8.352782047124195905396253404436, 9.677232815560667388698591909008

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