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Properties

Label 1-1480-1480.13-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.940 + 0.340i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
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  + (−0.342 − 0.939i)3-s + (−0.984 − 0.173i)7-s + (−0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (0.766 + 0.642i)13-s + (0.766 − 0.642i)17-s + (−0.342 − 0.939i)19-s + (0.173 + 0.984i)21-s + (0.5 − 0.866i)23-s + (0.866 + 0.5i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (−0.642 + 0.766i)33-s + (0.342 − 0.939i)39-s + (−0.766 − 0.642i)41-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)3-s + (−0.984 − 0.173i)7-s + (−0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (0.766 + 0.642i)13-s + (0.766 − 0.642i)17-s + (−0.342 − 0.939i)19-s + (0.173 + 0.984i)21-s + (0.5 − 0.866i)23-s + (0.866 + 0.5i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (−0.642 + 0.766i)33-s + (0.342 − 0.939i)39-s + (−0.766 − 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.940 + 0.340i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.940 + 0.340i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08892963038 - 0.5073607367i\)
\(L(\frac12)\) \(\approx\) \(-0.08892963038 - 0.5073607367i\)
\(L(1)\) \(\approx\) \(0.6461895986 - 0.3419282054i\)
\(L(1)\) \(\approx\) \(0.6461895986 - 0.3419282054i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.342 - 0.939i)T \)
7 \( 1 + (-0.984 - 0.173i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.766 + 0.642i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.984 - 0.173i)T \)
59 \( 1 + (-0.984 + 0.173i)T \)
61 \( 1 + (-0.642 + 0.766i)T \)
67 \( 1 + (-0.984 - 0.173i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.984 - 0.173i)T \)
83 \( 1 + (-0.642 - 0.766i)T \)
89 \( 1 + (-0.984 + 0.173i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−21.11227784033198808964238270908, −20.45109200041928717684413973603, −19.712065983867121881132332394579, −18.79210435314901781422556844652, −18.054592824845258729956969618682, −17.09831401335519673659962949276, −16.61047414196130490821920216352, −15.613084116428723956216590921885, −15.36399122250990278129553175495, −14.48476428951140994811904779048, −13.39544166589740396280884020493, −12.60906280259557324959779513304, −12.01638451119998788068347434150, −10.87693450404422633361001371863, −10.27376481699892533592376438269, −9.71466352786119353451048163285, −8.87147061864042833489375653522, −7.95607676859179205615621212313, −6.931555164423501366565790673240, −5.796326966189990788584164013613, −5.57077017417849938067861944282, −4.290714404616106571796390968770, −3.55008194516469600604521223478, −2.84613703434329749960276940895, −1.412983324818114623967924549, 0.22362975713383038969436920656, 1.18188024646426329571011190512, 2.50447714228291463058815926386, 3.145287873930070421035695878787, 4.3059313893957049516632263046, 5.55415313638840286433404331287, 6.08094612519667294401209543852, 6.95805999577039465818361306311, 7.541876789640209451385987757098, 8.67839483524233918986336524893, 9.20196122274091648976042673114, 10.4981520481250825374727225909, 11.05292004583386000510338390625, 11.91621262948823949176034540671, 12.707139728789906398148430472025, 13.48235065966659308707765524048, 13.76363003380344011958292599667, 14.90948237618811726740358648245, 16.01848185137444965066291030683, 16.52027149838453922423447632335, 17.13609373071954059405074209906, 18.252420764547323740161204503863, 18.84091419470250125159957972876, 19.14018789154564154953041081922, 20.175991074335854738064687657988

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