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Properties

Label 1-2520-2520.2477-r1-0-0
Degree $1$
Conductor $2520$
Sign $0.144 + 0.989i$
Analytic cond. $270.811$
Root an. cond. $270.811$
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  + (−0.5 − 0.866i)11-s + (0.866 + 0.5i)13-s + i·17-s − 19-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s − i·37-s + (−0.5 + 0.866i)41-s + (0.866 − 0.5i)43-s + (0.866 − 0.5i)47-s − i·53-s + (−0.5 + 0.866i)59-s + (−0.5 − 0.866i)61-s + (0.866 + 0.5i)67-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)11-s + (0.866 + 0.5i)13-s + i·17-s − 19-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s − i·37-s + (−0.5 + 0.866i)41-s + (0.866 − 0.5i)43-s + (0.866 − 0.5i)47-s − i·53-s + (−0.5 + 0.866i)59-s + (−0.5 − 0.866i)61-s + (0.866 + 0.5i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.144 + 0.989i$
Analytic conductor: \(270.811\)
Root analytic conductor: \(270.811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (2477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2520,\ (1:\ ),\ 0.144 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.022601806 + 0.8845172321i\)
\(L(\frac12)\) \(\approx\) \(1.022601806 + 0.8845172321i\)
\(L(1)\) \(\approx\) \(0.9785116285 + 0.03487530457i\)
\(L(1)\) \(\approx\) \(0.9785116285 + 0.03487530457i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + iT \)
19 \( 1 - T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 - iT \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + iT \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.06984479527378207145000354612, −18.31179419810566231247228300241, −17.71863760902997678273848573687, −17.12031306606183621987127979695, −16.053708575157253747876017380674, −15.59003583509590527771873529848, −14.99141850613361832439263723449, −13.858539702554020034106849920122, −13.55875916100032565408375649071, −12.499273727297016908742745569092, −12.06103168337653753482204648412, −11.05156385313841140814848204119, −10.3982765427520884838960568003, −9.72346202952234906951252633784, −8.84876736913742835049990777175, −8.05761751381647807212250006065, −7.40712005801783965994170588179, −6.47839985013969024453671158171, −5.78358654957115561415694308442, −4.81650094865254538901207582913, −4.1878415663925199079894973880, −3.12182542052028627716846321854, −2.34820776893175907008442746942, −1.36874679260848102211933633436, −0.27692496788028409473456350166, 0.7714478335740370730229838410, 1.82854633552082955892892172342, 2.66620615099766251936233343837, 3.76609700408563922383833489095, 4.24224701545000687703240025071, 5.42646215966361238583413181665, 6.1340143463890426535776392273, 6.69421440349264797812316804838, 7.89093045101942673033198887839, 8.44946920750564172617935070781, 9.03545792877365966382457139092, 10.18214788010309787028146883755, 10.73149838179801329168153879290, 11.35982215509265437430593144823, 12.31831883482998897759311865887, 12.97961312544881545205808204087, 13.708922144977309033853208319934, 14.361651744180131985577145596429, 15.186662674854103279784676160800, 15.948287522141260439253808617710, 16.52503233960070395622215907215, 17.23830740960394487852618941927, 18.10783235516745596740032158190, 18.75118014550251896587526438664, 19.30263348443055878407041932564

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