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Properties

Label 1-155-155.34-r1-0-0
Degree $1$
Conductor $155$
Sign $-0.807 + 0.590i$
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  + (−0.309 + 0.951i)2-s + (−0.978 − 0.207i)3-s + (−0.809 − 0.587i)4-s + (0.5 − 0.866i)6-s + (−0.913 + 0.406i)7-s + (0.809 − 0.587i)8-s + (0.913 + 0.406i)9-s + (0.104 − 0.994i)11-s + (0.669 + 0.743i)12-s + (0.669 − 0.743i)13-s + (−0.104 − 0.994i)14-s + (0.309 + 0.951i)16-s + (−0.104 − 0.994i)17-s + (−0.669 + 0.743i)18-s + (0.669 + 0.743i)19-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.978 − 0.207i)3-s + (−0.809 − 0.587i)4-s + (0.5 − 0.866i)6-s + (−0.913 + 0.406i)7-s + (0.809 − 0.587i)8-s + (0.913 + 0.406i)9-s + (0.104 − 0.994i)11-s + (0.669 + 0.743i)12-s + (0.669 − 0.743i)13-s + (−0.104 − 0.994i)14-s + (0.309 + 0.951i)16-s + (−0.104 − 0.994i)17-s + (−0.669 + 0.743i)18-s + (0.669 + 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $-0.807 + 0.590i$
Analytic conductor: \(16.6570\)
Root analytic conductor: \(16.6570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 155,\ (1:\ ),\ -0.807 + 0.590i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1461235929 + 0.4474864177i\)
\(L(\frac12)\) \(\approx\) \(0.1461235929 + 0.4474864177i\)
\(L(1)\) \(\approx\) \(0.5010900828 + 0.2121797292i\)
\(L(1)\) \(\approx\) \(0.5010900828 + 0.2121797292i\)

Euler product

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

−27.68141668879215775001976188472, −26.39748179427108576647350911340, −25.91377608904565512150938817561, −24.06109366945518333028133586030, −23.00424182550870592793983909837, −22.4292813593555523971730265797, −21.463329064399246693574689523228, −20.42373689023220956623400819442, −19.41320127875876243041245296171, −18.38087963650341498392473321494, −17.438394906549655998662507056731, −16.62048289839911833894456415739, −15.534516479830637980115006016552, −13.77788086214069735486981797678, −12.73714364120474197935871507945, −11.95760047654098525907448721794, −10.828305878265308136941221877123, −10.00501025830891599441948180915, −9.10134245089277070270385383154, −7.36085801798068787420405125667, −6.16406662930726400615497606794, −4.55082904963605085976708627874, −3.69163568693433538286895296, −1.82415594979114853568725020256, −0.285213840631325937884332354, 1.032495425545690999022202879339, 3.534888647934591484211553006656, 5.300148377599821040330528871050, 5.956803289139767447232746091113, 6.94723310285676042275406781910, 8.19311310395747606280075581113, 9.49407128008822527441195324043, 10.49326142076467882747290377521, 11.801988681678178257357158182766, 13.068281524159123822269358046, 13.90343136275868005250359007344, 15.58617031975032274224697608990, 16.12774421138763865993702813745, 16.93513759960895363378221653824, 18.31631706507664913533329642019, 18.55587160898860781590666164612, 19.92212743266272657820386157991, 21.71801471265133779420152596567, 22.543869187938617219768059013309, 23.19290586635628764771896255642, 24.311682465699293111151251180162, 25.035627482919626615725980281938, 26.067020759705353792700920136541, 27.25243500864373950049998655434, 27.87836021062126837252921265300

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