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Properties

Label 2-2520-2520.349-c0-0-6
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: $\chi_{2520} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ 0.906 + 0.422i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.713997800\)
\(L(\frac12)\) \(\approx\) \(1.713997800\)
\(L(1)\) 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

−8.720148061951405974865207631018, −8.209343205029825367802951004280, −7.18060499150458755221347984969, −6.49467152755708718690067313719, −6.05214632742501412084966754162, −5.27259412366754502764521110916, −4.53502743013843491449486782283, −3.44614095272100600026586522787, −2.39023254036715495282075204625, −1.04748939272848918861881423348, 1.47401105640868244903460212998, 2.79999614423067970579779723942, 3.46487482893327601816716318365, 4.28943054707681411587104664794, 5.38832264110189599690672321825, 5.88000831549521907766277393355, 6.56180701223905099246830315765, 7.03735156905605874684002436534, 8.873299467386999168602476580778, 9.364068553704311140177508296194

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