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Properties

Label 1-5160-5160.1109-r0-0-0
Degree $1$
Conductor $5160$
Sign $-0.817 + 0.575i$
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 + (−0.900 + 0.433i)11-s + (−0.988 − 0.149i)13-s + (0.365 − 0.930i)17-s + (0.826 − 0.563i)19-s + (0.0747 − 0.997i)23-s + (−0.955 + 0.294i)29-s + (−0.733 − 0.680i)31-s + (0.5 − 0.866i)37-s + (0.222 − 0.974i)41-s + (−0.900 − 0.433i)47-s + (−0.5 + 0.866i)49-s + (0.988 − 0.149i)53-s + (0.623 + 0.781i)59-s + (−0.733 + 0.680i)61-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (−0.900 + 0.433i)11-s + (−0.988 − 0.149i)13-s + (0.365 − 0.930i)17-s + (0.826 − 0.563i)19-s + (0.0747 − 0.997i)23-s + (−0.955 + 0.294i)29-s + (−0.733 − 0.680i)31-s + (0.5 − 0.866i)37-s + (0.222 − 0.974i)41-s + (−0.900 − 0.433i)47-s + (−0.5 + 0.866i)49-s + (0.988 − 0.149i)53-s + (0.623 + 0.781i)59-s + (−0.733 + 0.680i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5160\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 43\)
Sign: $-0.817 + 0.575i$
Analytic conductor: \(23.9629\)
Root analytic conductor: \(23.9629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5160} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5160,\ (0:\ ),\ -0.817 + 0.575i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08201400648 - 0.2591571112i\)
\(L(\frac12)\) \(\approx\) \(-0.08201400648 - 0.2591571112i\)
\(L(1)\) \(\approx\) \(0.7590488899 - 0.1929211400i\)
\(L(1)\) \(\approx\) \(0.7590488899 - 0.1929211400i\)

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 \( 1 + (-0.900 + 0.433i)T \)
13 \( 1 + (-0.988 - 0.149i)T \)
17 \( 1 + (0.365 - 0.930i)T \)
19 \( 1 + (0.826 - 0.563i)T \)
23 \( 1 + (0.0747 - 0.997i)T \)
29 \( 1 + (-0.955 + 0.294i)T \)
31 \( 1 + (-0.733 - 0.680i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (0.988 - 0.149i)T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (-0.733 + 0.680i)T \)
67 \( 1 + (0.826 - 0.563i)T \)
71 \( 1 + (0.0747 + 0.997i)T \)
73 \( 1 + (-0.988 - 0.149i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.955 - 0.294i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
97 \( 1 + (0.900 - 0.433i)T \)
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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

−18.4458339838918444433268101731, −17.840851311572732325437810085763, −16.95366980502696964690301059280, −16.39385619877603097056955118018, −15.747154771497656278408342351495, −15.06117376861784516505706233067, −14.5638263811407297102293194044, −13.67229497206424533853764404534, −12.90192805024810951251120275692, −12.53683955227728293572131547851, −11.664466683386903758790828020156, −11.162939321780314027421032740803, −10.05210539794060426102753772858, −9.7791834140152737347519205298, −8.96836344551628275124094901041, −8.11105501760222866223112775374, −7.63302329634266023294225136796, −6.76019790786579409199134971511, −5.78879080634208356168561861125, −5.505498127657223286974920668691, −4.67074728102257011376900035900, −3.51601616051371702110424660518, −3.064321182054795566361292402358, −2.17936969973678729099501299634, −1.377968735745965449885910801897, 0.08419179020808549096815258478, 0.85252778773825854338371388306, 2.17716288315849887637924209927, 2.746808271032752577388608844856, 3.5940914235075660636610537500, 4.44046937677735843546097037226, 5.1438637085780233222042028991, 5.74929499648845459851516582813, 6.92846484259908954327379322767, 7.34482771538324580942854883217, 7.76162062401704590081619606990, 8.951606700411583189316439411353, 9.569949620564032561488119562599, 10.1795265932948655083243972325, 10.743294388538370523528697249157, 11.58870978806197157000308796511, 12.30385153365311363469960852447, 13.10974958154098611264957256540, 13.37264238163574061092553851130, 14.44563577452460451161653296022, 14.749588122718410781950484377946, 15.80680299208244850933744646336, 16.2382512311892523756608103443, 16.89592418262959495442385393649, 17.56805216796658985009413074314

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