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Properties

Label 1-5160-5160.4379-r0-0-0
Degree $1$
Conductor $5160$
Sign $-0.216 + 0.976i$
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 − 11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s − 41-s − 47-s + (−0.5 + 0.866i)49-s + (0.5 − 0.866i)53-s − 59-s + (0.5 + 0.866i)61-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s − 11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s − 41-s − 47-s + (−0.5 + 0.866i)49-s + (0.5 − 0.866i)53-s − 59-s + (0.5 + 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5160\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 43\)
Sign: $-0.216 + 0.976i$
Analytic conductor: \(23.9629\)
Root analytic conductor: \(23.9629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5160} (4379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5160,\ (0:\ ),\ -0.216 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02268389506 + 0.02827855703i\)
\(L(\frac12)\) \(\approx\) \(0.02268389506 + 0.02827855703i\)
\(L(1)\) \(\approx\) \(0.7164523376 - 0.1793113603i\)
\(L(1)\) \(\approx\) \(0.7164523376 - 0.1793113603i\)

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 \( 1 - T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 - T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 - T \)
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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

−17.78098010872531910549411855893, −17.22566308189073341612494377466, −16.38069301022772707966113415957, −15.75160883854338598199166689817, −15.23328205244820855386433683138, −14.64038900818165744443638487638, −13.69688250752796093613680805892, −13.05730380729585430861168001643, −12.555800988056603792904481863921, −11.83061863274529030177797381216, −11.061687014813144027495160016672, −10.45488622903551960666119215243, −9.62272920836046180650925773678, −8.9361004083330533759124967672, −8.508105332140707055204961271271, −7.495569950089913683566903479768, −6.81530179577782298679224067795, −6.19096971529907448593868128357, −5.236090748593943254451208915905, −4.85850139909975942311505409144, −3.76855129325516004311137000487, −2.9941757666294066406538993971, −2.25877949402192956088587267679, −1.579287185202247439156718093586, −0.012187791932536073175935945399, 0.72229168564383499264941941222, 1.99129739410619015976558523724, 2.78329525292170079783311782078, 3.421394039977421405084175297122, 4.38959050832090223458720564604, 4.973457451872265804860732099471, 5.81317151797428036388293200688, 6.62642736186653856315822397319, 7.263862659912772322372198624805, 8.0077329407422346426385380130, 8.499242431987478524985568697958, 9.79192833677877412064249421630, 9.94963907189711689189462316801, 10.72954560494063257258387826468, 11.37446022439010810269080682866, 12.32847643903178181135239134393, 12.92314206165241516758826196098, 13.47766209093866751302030573437, 14.03935145938005287520103223382, 15.16270540474722553996562568147, 15.30463611269836456932917619665, 16.385931941200691064594347421407, 16.75113761819373048546088630412, 17.47053290270129714631477125047, 18.220344621889781475072351344524

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