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Properties

Label 1-1275-1275.614-r1-0-0
Degree $1$
Conductor $1275$
Sign $-0.972 + 0.232i$
Analytic cond. $137.017$
Root an. cond. $137.017$
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  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (0.707 + 0.707i)7-s + (0.587 + 0.809i)8-s + (−0.891 + 0.453i)11-s + (0.309 + 0.951i)13-s + (0.453 + 0.891i)14-s + (0.309 + 0.951i)16-s + (−0.587 − 0.809i)19-s + (−0.987 + 0.156i)22-s + (0.891 − 0.453i)23-s + i·26-s + (0.156 + 0.987i)28-s + (−0.987 + 0.156i)29-s + (−0.156 + 0.987i)31-s + i·32-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (0.707 + 0.707i)7-s + (0.587 + 0.809i)8-s + (−0.891 + 0.453i)11-s + (0.309 + 0.951i)13-s + (0.453 + 0.891i)14-s + (0.309 + 0.951i)16-s + (−0.587 − 0.809i)19-s + (−0.987 + 0.156i)22-s + (0.891 − 0.453i)23-s + i·26-s + (0.156 + 0.987i)28-s + (−0.987 + 0.156i)29-s + (−0.156 + 0.987i)31-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1275\)    =    \(3 \cdot 5^{2} \cdot 17\)
Sign: $-0.972 + 0.232i$
Analytic conductor: \(137.017\)
Root analytic conductor: \(137.017\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1275} (614, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1275,\ (1:\ ),\ -0.972 + 0.232i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3719004761 + 3.150181198i\)
\(L(\frac12)\) \(\approx\) \(0.3719004761 + 3.150181198i\)
\(L(1)\) \(\approx\) \(1.582207815 + 0.9103737764i\)
\(L(1)\) \(\approx\) \(1.582207815 + 0.9103737764i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (-0.891 + 0.453i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.587 - 0.809i)T \)
23 \( 1 + (0.891 - 0.453i)T \)
29 \( 1 + (-0.987 + 0.156i)T \)
31 \( 1 + (-0.156 + 0.987i)T \)
37 \( 1 + (-0.891 - 0.453i)T \)
41 \( 1 + (-0.453 + 0.891i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (-0.951 + 0.309i)T \)
61 \( 1 + (-0.891 + 0.453i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (0.987 - 0.156i)T \)
73 \( 1 + (-0.453 - 0.891i)T \)
79 \( 1 + (-0.156 - 0.987i)T \)
83 \( 1 + (-0.587 - 0.809i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.987 + 0.156i)T \)
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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

−20.594790760547848664086991273528, −20.13794967137416997928584391040, −18.93124948887359847189516823647, −18.501968031088369547994416267386, −17.20356116122600278597422164077, −16.71795815511689420025928814351, −15.409318675744866346964629160513, −15.26099767159370166373322119448, −14.11484215667248572788114557099, −13.53797551361961617466563770876, −12.88284986999612770158454548969, −12.04019653219123743420216708891, −10.89851241792549532858056254299, −10.7779648922484025231519994757, −9.826693235761289614076538771525, −8.44533333533172861454986616518, −7.66840989139152511290120524, −6.90706588933351982501913221977, −5.60392206451154351155860227284, −5.347928619426189732021779541362, −4.12996737338058166936828066488, −3.49314857932661984981911697187, −2.43936394230381938332098880302, −1.44530477688536773669606958834, −0.38280884281577260550983224269, 1.60049884334685975485752933507, 2.36488215812257467580868821095, 3.26758570140446977390589224181, 4.55019195322749648538392471417, 4.92640940389654082492437015649, 5.87798728023573974288937330162, 6.79585458660322641252551568559, 7.53777343408119442645991353400, 8.49206522590019919884923190086, 9.18749209045578028125974112853, 10.6592415915704488331480076393, 11.159431063280265679568611551890, 12.03689913393812625790906844298, 12.760092655028360717703000807, 13.45900982485355069079297042775, 14.3850551671287607038854268530, 14.999752858629025744705043120820, 15.62693275546550207651625609609, 16.42394714163289790561124330532, 17.27075061791332267653704342874, 18.08595514431910991689554230679, 18.877319350535624122422283843454, 19.86477764032214609970432488201, 20.79110064111301023642092020299, 21.29902907063142064897934843721

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