Properties

Label 1-151-151.47-r0-0-0
Degree $1$
Conductor $151$
Sign $-0.415 + 0.909i$
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  + (0.669 + 0.743i)2-s + (0.968 + 0.248i)3-s + (−0.104 + 0.994i)4-s + (−0.855 + 0.518i)5-s + (0.463 + 0.886i)6-s + (−0.756 − 0.653i)7-s + (−0.809 + 0.587i)8-s + (0.876 + 0.481i)9-s + (−0.957 − 0.289i)10-s + (−0.268 + 0.963i)11-s + (−0.348 + 0.937i)12-s + (0.783 + 0.621i)13-s + (−0.0209 − 0.999i)14-s + (−0.957 + 0.289i)15-s + (−0.978 − 0.207i)16-s + (0.944 + 0.328i)17-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (0.968 + 0.248i)3-s + (−0.104 + 0.994i)4-s + (−0.855 + 0.518i)5-s + (0.463 + 0.886i)6-s + (−0.756 − 0.653i)7-s + (−0.809 + 0.587i)8-s + (0.876 + 0.481i)9-s + (−0.957 − 0.289i)10-s + (−0.268 + 0.963i)11-s + (−0.348 + 0.937i)12-s + (0.783 + 0.621i)13-s + (−0.0209 − 0.999i)14-s + (−0.957 + 0.289i)15-s + (−0.978 − 0.207i)16-s + (0.944 + 0.328i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $-0.415 + 0.909i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ -0.415 + 0.909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9068857803 + 1.411573620i\)
\(L(\frac12)\) \(\approx\) \(0.9068857803 + 1.411573620i\)
\(L(1)\) \(\approx\) \(1.229183774 + 0.9620937477i\)
\(L(1)\) \(\approx\) \(1.229183774 + 0.9620937477i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (0.669 + 0.743i)T \)
3 \( 1 + (0.968 + 0.248i)T \)
5 \( 1 + (-0.855 + 0.518i)T \)
7 \( 1 + (-0.756 - 0.653i)T \)
11 \( 1 + (-0.268 + 0.963i)T \)
13 \( 1 + (0.783 + 0.621i)T \)
17 \( 1 + (0.944 + 0.328i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.913 - 0.406i)T \)
29 \( 1 + (0.0627 - 0.998i)T \)
31 \( 1 + (0.604 - 0.796i)T \)
37 \( 1 + (0.146 - 0.989i)T \)
41 \( 1 + (0.535 - 0.844i)T \)
43 \( 1 + (-0.756 + 0.653i)T \)
47 \( 1 + (-0.999 - 0.0418i)T \)
53 \( 1 + (-0.637 + 0.770i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (-0.699 - 0.714i)T \)
67 \( 1 + (0.876 - 0.481i)T \)
71 \( 1 + (0.944 - 0.328i)T \)
73 \( 1 + (-0.187 + 0.982i)T \)
79 \( 1 + (-0.992 + 0.125i)T \)
83 \( 1 + (-0.425 + 0.904i)T \)
89 \( 1 + (0.996 - 0.0836i)T \)
97 \( 1 + (-0.0209 + 0.999i)T \)
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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

−27.747064825672072170370023848142, −27.06695373845937261947289731667, −25.5356325142389460023457533586, −24.78390777933238973298831828332, −23.65380484513160028864503292476, −22.98369004247506696810093770227, −21.53553660188532690477483737452, −20.83551962498309151102483349509, −19.837899182035809397340955777801, −19.03247245099324180840215878149, −18.52328394618785112661071953629, −16.23476028026283953285648968278, −15.473920926209527680479334364898, −14.4962683869559282286359640204, −13.20561601230763778337893656406, −12.69801113090871638091037142663, −11.60286303141577591760064085925, −10.24495498304996481331496735462, −8.97633057862039544038602656934, −8.17657507846310875471955882278, −6.47195730882199940014944102005, −5.11506281754459391009469114635, −3.47997232092356230723039992875, −3.08934910846040420103923480590, −1.20277434799558850958564721180, 2.66642879735267794832654523014, 3.78937698897102550395530467657, 4.46741747347173207516622312548, 6.50039068998503803564635479459, 7.35484255263644748373341784777, 8.25156982129878285718800096561, 9.56759449543155841870997760016, 10.92290036303000691079999292043, 12.47546464621644823903343874540, 13.336059384661801555691981571738, 14.404386270935941253300649359146, 15.23019424918544597347887854646, 15.95605914783265464881870210406, 17.00656031009008108401297088568, 18.61148200742925879838278410667, 19.50916381833465026847501750912, 20.62382608952211514004363864453, 21.46645930558415065319335005171, 22.94480969304744796305369427508, 23.20663774339076207509214843819, 24.468740507976134513970200389305, 25.85313605553523605616684717757, 25.97739587783901564835743259794, 26.945091638326249988478614808057, 28.120732609008625312651388996257

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