Properties

Label 2-2520-5.4-c1-0-27
Degree $2$
Conductor $2520$
Sign $0.783 + 0.621i$
Analytic cond. $20.1223$
Root an. cond. $4.48578$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.75 + 1.38i)5-s i·7-s − 5.14·11-s − 4.64i·13-s + 3.86i·17-s + 0.778·19-s − 5.00i·23-s + (1.14 + 4.86i)25-s + 9.42·29-s + 4.72·31-s + (1.38 − 1.75i)35-s − 6i·37-s + 1.00·41-s − 7.00i·43-s − 11.4i·47-s + ⋯
L(s)  = 1  + (0.783 + 0.621i)5-s − 0.377i·7-s − 1.55·11-s − 1.28i·13-s + 0.938i·17-s + 0.178·19-s − 1.04i·23-s + (0.228 + 0.973i)25-s + 1.74·29-s + 0.848·31-s + (0.234 − 0.296i)35-s − 0.986i·37-s + 0.157·41-s − 1.06i·43-s − 1.66i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.783 + 0.621i$
Analytic conductor: \(20.1223\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ 0.783 + 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.806989690\)
\(L(\frac12)\) \(\approx\) \(1.806989690\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-1.75 - 1.38i)T \)
7 \( 1 + iT \)
good11 \( 1 + 5.14T + 11T^{2} \)
13 \( 1 + 4.64iT - 13T^{2} \)
17 \( 1 - 3.86iT - 17T^{2} \)
19 \( 1 - 0.778T + 19T^{2} \)
23 \( 1 + 5.00iT - 23T^{2} \)
29 \( 1 - 9.42T + 29T^{2} \)
31 \( 1 - 4.72T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 1.00T + 41T^{2} \)
43 \( 1 + 7.00iT - 43T^{2} \)
47 \( 1 + 11.4iT - 47T^{2} \)
53 \( 1 + 7.55iT - 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 - 11.7iT - 67T^{2} \)
71 \( 1 + 2.72T + 71T^{2} \)
73 \( 1 - 5.00iT - 73T^{2} \)
79 \( 1 + 5.68T + 79T^{2} \)
83 \( 1 - 4.67iT - 83T^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 - 1.58iT - 97T^{2} \)
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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

−8.530601551096776172119608860710, −8.236343278901774575263465065883, −7.22552526780389756924294943919, −6.56535488839966997246697963980, −5.59231192321348650903620388487, −5.16841492202844994201900702739, −3.91392252897854704702633376272, −2.85139326443831076600829781629, −2.27923785208909139433265285932, −0.67034462645585601602404197236, 1.11373875470825605196781497375, 2.31992878004315136132630275793, 2.99053629160952327448134668438, 4.59994488451215588399093542221, 4.92943253575542732332143446993, 5.86322420158536554422306084464, 6.57133296460824357482357601348, 7.55242805150135453154945149033, 8.309380910320847275279505237170, 9.021980896498859444710872882651

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