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Properties

Label 1-5160-5160.2309-r0-0-0
Degree $1$
Conductor $5160$
Sign $0.626 - 0.779i$
Analytic cond. $23.9629$
Root an. cond. $23.9629$
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)7-s + (0.623 − 0.781i)11-s + (−0.733 + 0.680i)13-s + (0.955 − 0.294i)17-s + (−0.988 + 0.149i)19-s + (0.365 + 0.930i)23-s + (−0.0747 − 0.997i)29-s + (0.826 − 0.563i)31-s + (0.5 − 0.866i)37-s + (0.900 + 0.433i)41-s + (0.623 + 0.781i)47-s + (−0.5 + 0.866i)49-s + (0.733 + 0.680i)53-s + (−0.222 + 0.974i)59-s + (0.826 + 0.563i)61-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (0.623 − 0.781i)11-s + (−0.733 + 0.680i)13-s + (0.955 − 0.294i)17-s + (−0.988 + 0.149i)19-s + (0.365 + 0.930i)23-s + (−0.0747 − 0.997i)29-s + (0.826 − 0.563i)31-s + (0.5 − 0.866i)37-s + (0.900 + 0.433i)41-s + (0.623 + 0.781i)47-s + (−0.5 + 0.866i)49-s + (0.733 + 0.680i)53-s + (−0.222 + 0.974i)59-s + (0.826 + 0.563i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5160\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 43\)
Sign: $0.626 - 0.779i$
Analytic conductor: \(23.9629\)
Root analytic conductor: \(23.9629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5160} (2309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5160,\ (0:\ ),\ 0.626 - 0.779i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.450083702 - 0.6948514646i\)
\(L(\frac12)\) \(\approx\) \(1.450083702 - 0.6948514646i\)
\(L(1)\) \(\approx\) \(1.033402337 - 0.1620372863i\)
\(L(1)\) \(\approx\) \(1.033402337 - 0.1620372863i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
43 \( 1 \)
good7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.623 - 0.781i)T \)
13 \( 1 + (-0.733 + 0.680i)T \)
17 \( 1 + (0.955 - 0.294i)T \)
19 \( 1 + (-0.988 + 0.149i)T \)
23 \( 1 + (0.365 + 0.930i)T \)
29 \( 1 + (-0.0747 - 0.997i)T \)
31 \( 1 + (0.826 - 0.563i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (0.733 + 0.680i)T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + (0.826 + 0.563i)T \)
67 \( 1 + (-0.988 + 0.149i)T \)
71 \( 1 + (0.365 - 0.930i)T \)
73 \( 1 + (-0.733 + 0.680i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.0747 + 0.997i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
97 \( 1 + (-0.623 + 0.781i)T \)
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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

−18.11220773334622712491034452396, −17.33084786107298300131831135047, −16.8738064103317985708782739318, −16.08310021960164433318340168181, −15.333054937677122894052197698187, −14.72420113606631341503011742765, −14.43162500741280047821804309685, −13.2286276566388265282393332181, −12.52746466630554266233945735726, −12.33126726361267721671868598951, −11.538412445328619183239849687893, −10.47737319171688012020686451793, −10.0736476191789562647643720472, −9.296471461129166315271743636660, −8.63102859485819967133341515213, −7.985691748742947356547484663975, −6.99597093042858106942168215751, −6.53468526686364129714281075091, −5.66646001405956577249707162356, −5.00125109464933615140905775110, −4.24365893076464880611284068743, −3.28395160273746027026686210921, −2.61099425343558327940294764965, −1.88103512808175517077932776865, −0.78071233596559662415236633796, 0.58157064017643707975635697707, 1.346862767642720480917082765602, 2.46821280956159928863967981232, 3.18986943218106817031569984194, 4.15996299415611818742938731666, 4.38812030013033015300233659745, 5.75910311828310102419632588251, 6.096183508236290081905280462357, 7.10592338262776346825976117231, 7.51267177217242184468336579983, 8.37500179514191358477829206189, 9.305835386311406800944268013949, 9.69764437472582823133298431990, 10.48706320930765184430780682553, 11.221892892247079048941734772489, 11.84812002918988683962331016182, 12.54112759408638956402546080413, 13.3978986564352871759956315082, 13.84561033475637598505833226555, 14.53448314599270851627053156766, 15.14778835410665242899023398409, 16.153816736137751523208707438616, 16.61135064811936003648375348998, 17.12396143779175660625716216026, 17.69301119118804921284401597583

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