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Properties

Label 1-155-155.68-r0-0-0
Degree $1$
Conductor $155$
Sign $0.952 - 0.303i$
Analytic cond. $0.719816$
Root an. cond. $0.719816$
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  − i·2-s + (0.866 + 0.5i)3-s − 4-s + (0.5 − 0.866i)6-s + (0.866 + 0.5i)7-s + i·8-s + (0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.866 − 0.5i)12-s + (−0.866 + 0.5i)13-s + (0.5 − 0.866i)14-s + 16-s + (−0.866 − 0.5i)17-s + (0.866 − 0.5i)18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  − i·2-s + (0.866 + 0.5i)3-s − 4-s + (0.5 − 0.866i)6-s + (0.866 + 0.5i)7-s + i·8-s + (0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.866 − 0.5i)12-s + (−0.866 + 0.5i)13-s + (0.5 − 0.866i)14-s + 16-s + (−0.866 − 0.5i)17-s + (0.866 − 0.5i)18-s + (0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $0.952 - 0.303i$
Analytic conductor: \(0.719816\)
Root analytic conductor: \(0.719816\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 155,\ (0:\ ),\ 0.952 - 0.303i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.384735179 - 0.2153953581i\)
\(L(\frac12)\) \(\approx\) \(1.384735179 - 0.2153953581i\)
\(L(1)\) \(\approx\) \(1.275623829 - 0.2482123964i\)
\(L(1)\) \(\approx\) \(1.275623829 - 0.2482123964i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 - iT \)
29 \( 1 + T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + T \)
97 \( 1 - iT \)
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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

−27.22038358907873850731962356703, −27.09816934697276348377500122981, −25.95062209492277893623957873989, −24.87884195716201029470188001912, −24.333865746519889596098264622766, −23.57670743781355928989512222302, −22.23669086519236403951989399651, −21.17793785772452639668131240646, −19.89115451158442001314864282153, −19.0445337281189655897687632765, −17.8410749951670587342252811018, −17.20219295615527690820698497823, −15.814352714490160177848624937362, −14.75490067051213904339931251064, −14.07198670665884101468217780230, −13.2580287845685882736995441359, −11.94704738690629694634932400175, −10.22967961951001752776625042643, −8.9590954357669297018415371980, −8.06663505394812391584558774442, −7.2921844433783304680651296817, −6.05297603163107163584012700173, −4.58194163859864987880584622774, −3.34631579517164070829320733671, −1.35366039423925430863145985634, 1.891759481036663149965766626408, 2.724559293853303962782644573659, 4.38059710385770507004388702121, 4.91395691230648760349982751385, 7.26049651307239309782296035651, 8.65146396081426653408536713171, 9.29268284981624669171087586680, 10.38942552261010027129442608733, 11.55484940977547981955801193439, 12.522447960500366269254934152, 13.898262348266327151815575172751, 14.54718795503386951911794613283, 15.58354016004115396781172471592, 17.24451810022372779591897579372, 18.162320126859657262311260483481, 19.327856419026274456531560307, 20.11866827162021339283910630578, 20.88624991556497472286916704343, 21.85543134828396445376186242963, 22.49270232539345054788552903656, 24.11374262706049025063257571120, 24.997628268206646934813494156, 26.34662769630878258809482981047, 27.00784122650710114322138091878, 27.90648760606051783389640393980

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