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Properties

Label 1-151-151.19-r0-0-0
Degree $1$
Conductor $151$
Sign $0.914 + 0.405i$
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  + 2-s + (−0.809 − 0.587i)3-s + 4-s + (0.309 + 0.951i)5-s + (−0.809 − 0.587i)6-s + (0.309 + 0.951i)7-s + 8-s + (0.309 + 0.951i)9-s + (0.309 + 0.951i)10-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)15-s + 16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + 2-s + (−0.809 − 0.587i)3-s + 4-s + (0.309 + 0.951i)5-s + (−0.809 − 0.587i)6-s + (0.309 + 0.951i)7-s + 8-s + (0.309 + 0.951i)9-s + (0.309 + 0.951i)10-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)15-s + 16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.914 + 0.405i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.914 + 0.405i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.625167812 + 0.3438982349i\)
\(L(\frac12)\) \(\approx\) \(1.625167812 + 0.3438982349i\)
\(L(1)\) \(\approx\) \(1.529480886 + 0.1432956876i\)
\(L(1)\) \(\approx\) \(1.529480886 + 0.1432956876i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.309 - 0.951i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
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

−28.37829286068103504155851754511, −27.00892978580089016937860247507, −26.11627037865456132844262130093, −24.53171588409296950098681110555, −23.932357991782745224862683567, −23.20188457620004957153522398283, −22.0332623445859340735645746044, −21.11351586657249877311602618695, −20.66671500267449830133308358159, −19.38546149740984048731369554378, −17.516454805261177145403288207368, −16.72268543923977707103341046029, −16.0978848972220728335995031576, −14.85507504777565907712254905502, −13.66833586452623784299373099218, −12.74906972855538047668046352911, −11.69821732375317636134357943620, −10.70793779395990317492084631763, −9.705743724369505776177246163760, −7.909552888172171377285256539517, −6.547240706042442099036428690803, −5.254475343291599648721255916412, −4.72420120220711212016198433016, −3.449974472802786041525249129686, −1.351459428059249416250434268290, 2.05039521372085632206144421970, 2.95542132362548342027284575463, 5.143166730818015448228034219, 5.52576163301927102815966152524, 6.990523817173102610417477164815, 7.604019949819899246995572584958, 9.9254707472731971876229263043, 11.02195681232530988139090894381, 11.895305860311378349763253725572, 12.77066145411376153124519084369, 13.84967522063748483750852423224, 14.97745201261285215823274241115, 15.76914740532553417116519119077, 17.19804111383921951695808100537, 18.23432808898274066298648990285, 18.97299838704618042509504143154, 20.49151195955203213123680171810, 21.60115729886469492061397284339, 22.51432425980519117051713693477, 22.86890557361096442018366054987, 24.18577577765335781782125135413, 24.91480706016248338184508237429, 25.773578661449915962926575078, 27.29992962882111278813221746533, 28.55885547195670187147099307016

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