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Properties

Label 1-155-155.79-r1-0-0
Degree $1$
Conductor $155$
Sign $0.571 + 0.820i$
Analytic cond. $16.6570$
Root an. cond. $16.6570$
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  + (0.809 + 0.587i)2-s + (−0.104 − 0.994i)3-s + (0.309 + 0.951i)4-s + (0.5 − 0.866i)6-s + (0.978 + 0.207i)7-s + (−0.309 + 0.951i)8-s + (−0.978 + 0.207i)9-s + (−0.669 + 0.743i)11-s + (0.913 − 0.406i)12-s + (0.913 + 0.406i)13-s + (0.669 + 0.743i)14-s + (−0.809 + 0.587i)16-s + (0.669 + 0.743i)17-s + (−0.913 − 0.406i)18-s + (0.913 − 0.406i)19-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (−0.104 − 0.994i)3-s + (0.309 + 0.951i)4-s + (0.5 − 0.866i)6-s + (0.978 + 0.207i)7-s + (−0.309 + 0.951i)8-s + (−0.978 + 0.207i)9-s + (−0.669 + 0.743i)11-s + (0.913 − 0.406i)12-s + (0.913 + 0.406i)13-s + (0.669 + 0.743i)14-s + (−0.809 + 0.587i)16-s + (0.669 + 0.743i)17-s + (−0.913 − 0.406i)18-s + (0.913 − 0.406i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $0.571 + 0.820i$
Analytic conductor: \(16.6570\)
Root analytic conductor: \(16.6570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 155,\ (1:\ ),\ 0.571 + 0.820i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.648402080 + 1.382928164i\)
\(L(\frac12)\) \(\approx\) \(2.648402080 + 1.382928164i\)
\(L(1)\) \(\approx\) \(1.727773626 + 0.4576636120i\)
\(L(1)\) \(\approx\) \(1.727773626 + 0.4576636120i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (-0.104 - 0.994i)T \)
7 \( 1 + (0.978 + 0.207i)T \)
11 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (0.913 - 0.406i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.978 + 0.207i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 - T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (0.669 - 0.743i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (-0.104 + 0.994i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−27.55699792341144755141139645363, −26.99522000665732921182395591588, −25.57435929254292177565170574779, −24.3845725256787122959676267828, −23.32861003107008798043393932111, −22.65746952358855097408220969830, −21.328405784820108689393706610859, −21.00479965276653315635661532297, −20.15601118269551227686495682677, −18.78251254864175575082550137679, −17.64791261808391221396860350890, −16.11169358608543085187694790999, −15.51865829811285105365188979039, −14.22306462753917557925397628871, −13.683951344929273988026162251324, −12.02829446215215439284354827124, −11.11309188498665471839638209147, −10.45900871630675588993529028122, −9.20944871018824466726922154373, −7.7818351087127423235257846417, −5.78102258572393939749762866775, −5.1613458427766369962368346035, −3.88614555323208804897746534330, −2.88150689603967310420901837227, −0.97551625182349902546201648975, 1.55725143592449258235705179335, 2.91533131174302397380040192304, 4.646084332473726640845366424, 5.6706972977681712212917112169, 6.82636107517490052307651574777, 7.8413677555863021575271178883, 8.63250109630376355192973237955, 10.84220457959949333862373465926, 11.899862489444272488066351579649, 12.70718510592598736890405283259, 13.77664646028598205521428850212, 14.557122939118911990581162785785, 15.67121679028792970469633432187, 16.94034787516989165431266198439, 17.86532522108367731165044506490, 18.62068502305556402338010155910, 20.26053563274744374520120212027, 21.00509832903496244896985453436, 22.20683353594719204276654218623, 23.403314561791018855252247173235, 23.77048943183318388845089288317, 24.79165742455715496558574912066, 25.58674878673672150220674035564, 26.49978543383224573120825953176, 28.06685489230783767431810085494

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