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Properties

Label 1-151-151.16-r0-0-0
Degree $1$
Conductor $151$
Sign $0.718 + 0.695i$
Analytic cond. $0.701241$
Root an. cond. $0.701241$
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)2-s + (0.309 + 0.951i)3-s + (−0.5 + 0.866i)4-s + (0.913 + 0.406i)5-s + (0.669 − 0.743i)6-s + (0.913 + 0.406i)7-s + 8-s + (−0.809 + 0.587i)9-s + (−0.104 − 0.994i)10-s + (−0.978 + 0.207i)11-s + (−0.978 − 0.207i)12-s + (−0.978 − 0.207i)13-s + (−0.104 − 0.994i)14-s + (−0.104 + 0.994i)15-s + (−0.5 − 0.866i)16-s + (−0.104 + 0.994i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.309 + 0.951i)3-s + (−0.5 + 0.866i)4-s + (0.913 + 0.406i)5-s + (0.669 − 0.743i)6-s + (0.913 + 0.406i)7-s + 8-s + (−0.809 + 0.587i)9-s + (−0.104 − 0.994i)10-s + (−0.978 + 0.207i)11-s + (−0.978 − 0.207i)12-s + (−0.978 − 0.207i)13-s + (−0.104 − 0.994i)14-s + (−0.104 + 0.994i)15-s + (−0.5 − 0.866i)16-s + (−0.104 + 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.718 + 0.695i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.718 + 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9485387504 + 0.3840266291i\)
\(L(\frac12)\) \(\approx\) \(0.9485387504 + 0.3840266291i\)
\(L(1)\) \(\approx\) \(0.9782981276 + 0.1396425867i\)
\(L(1)\) \(\approx\) \(0.9782981276 + 0.1396425867i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.913 + 0.406i)T \)
7 \( 1 + (0.913 + 0.406i)T \)
11 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (-0.978 - 0.207i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.913 - 0.406i)T \)
47 \( 1 + (-0.978 - 0.207i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + T \)
61 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (-0.104 + 0.994i)T \)
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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

−27.83765063029722194790643493183, −26.68672689911942924682366531921, −25.87735480251590572169200735811, −24.88290560813999042185236228340, −24.23099828824816464053910134842, −23.64422728418754409772446829063, −22.238290327866982676086143285138, −20.75523280568687162305058764552, −19.927627431339467224047967871749, −18.56655732057592307340010135775, −17.88962586353777593248637859363, −17.23374004877085426367105145595, −16.0593650532664426579970080093, −14.55687037127533019025903311422, −13.90159382082147693502231573080, −13.09814664098868737599138984235, −11.53909451401619136413204972533, −10.030347738190336513785175001000, −9.03692630733735677227247116802, −7.833462533807940936027356320923, −7.21176641624443957044867419868, −5.73724053319094865159027543391, −4.91323293859033229961643881655, −2.37513408860326009310765926952, −1.08386487596580787896095365212, 2.07060862234763035904104960334, 2.867676159812801026426930349969, 4.50975609265195429798620384034, 5.50693337536113081211522080550, 7.68634499183297301260435183703, 8.670912434410147731547905988969, 9.881549540258472229945492740523, 10.40260356275019582957062246677, 11.46495340917952803745416664726, 12.81477659236762199831171860528, 14.07756431955136307498394608199, 14.9071189803979032731337696346, 16.31025871731909568055367420340, 17.54778677677773908507106670019, 18.0590264366649765287158474424, 19.39779065893836951817944123912, 20.505306953166906828234059049169, 21.26415113550416795893819698394, 21.82569844920714233274541327505, 22.75100155077244366911334610295, 24.55869823826422484665802202200, 25.64358024903248179515507668335, 26.53718657057912367668438956984, 27.068116597140917581385006246587, 28.43517399065872388815520738520

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