Properties

Label 1-151-151.88-r0-0-0
Degree $1$
Conductor $151$
Sign $0.730 - 0.682i$
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 + (0.535 + 0.844i)3-s + (−0.104 − 0.994i)4-s + (0.996 − 0.0836i)5-s + (0.985 + 0.166i)6-s + (0.228 − 0.973i)7-s + (−0.809 − 0.587i)8-s + (−0.425 + 0.904i)9-s + (0.604 − 0.796i)10-s + (−0.999 − 0.0418i)11-s + (0.783 − 0.621i)12-s + (0.832 + 0.553i)13-s + (−0.570 − 0.821i)14-s + (0.604 + 0.796i)15-s + (−0.978 + 0.207i)16-s + (−0.957 − 0.289i)17-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + (0.535 + 0.844i)3-s + (−0.104 − 0.994i)4-s + (0.996 − 0.0836i)5-s + (0.985 + 0.166i)6-s + (0.228 − 0.973i)7-s + (−0.809 − 0.587i)8-s + (−0.425 + 0.904i)9-s + (0.604 − 0.796i)10-s + (−0.999 − 0.0418i)11-s + (0.783 − 0.621i)12-s + (0.832 + 0.553i)13-s + (−0.570 − 0.821i)14-s + (0.604 + 0.796i)15-s + (−0.978 + 0.207i)16-s + (−0.957 − 0.289i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.730 - 0.682i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.730 - 0.682i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.802617031 - 0.7112435940i\)
\(L(\frac12)\) \(\approx\) \(1.802617031 - 0.7112435940i\)
\(L(1)\) \(\approx\) \(1.684299444 - 0.4783871634i\)
\(L(1)\) \(\approx\) \(1.684299444 - 0.4783871634i\)

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 \( 1 + (0.535 + 0.844i)T \)
5 \( 1 + (0.996 - 0.0836i)T \)
7 \( 1 + (0.228 - 0.973i)T \)
11 \( 1 + (-0.999 - 0.0418i)T \)
13 \( 1 + (0.832 + 0.553i)T \)
17 \( 1 + (-0.957 - 0.289i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.913 + 0.406i)T \)
29 \( 1 + (0.968 + 0.248i)T \)
31 \( 1 + (-0.0209 - 0.999i)T \)
37 \( 1 + (-0.895 + 0.444i)T \)
41 \( 1 + (-0.637 + 0.770i)T \)
43 \( 1 + (0.228 + 0.973i)T \)
47 \( 1 + (-0.348 - 0.937i)T \)
53 \( 1 + (-0.929 + 0.368i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.463 + 0.886i)T \)
67 \( 1 + (-0.425 - 0.904i)T \)
71 \( 1 + (-0.957 + 0.289i)T \)
73 \( 1 + (0.728 + 0.684i)T \)
79 \( 1 + (0.876 - 0.481i)T \)
83 \( 1 + (-0.187 + 0.982i)T \)
89 \( 1 + (-0.756 - 0.653i)T \)
97 \( 1 + (-0.570 + 0.821i)T \)
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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

−28.474318126349731922755780459916, −26.73091283968935355863986162455, −25.65047026399276118469389011838, −25.32674841041422132710914819609, −24.381512272469875305052319950297, −23.53333744950746778841527302562, −22.3817038904287360100396237983, −21.25994733870028730310183218094, −20.69338764337312031391882672592, −18.988345947827781587276397519161, −17.890791641064110506672229207084, −17.54446824679148857713470410792, −15.75669629378736068323927157958, −15.00638374788737406401338104757, −13.886960988060181830679468446652, −13.091037088645009843634719434111, −12.44827051010062679270066791463, −10.8439730238298674839247625637, −8.87890756640928383701248410967, −8.42361784084175121486364640611, −6.90327549830919518910192066928, −6.04826368841372746365364103162, −5.01849184824062360431433771694, −3.01953149310061114486964208704, −2.14837969965052694354040358329, 1.736746027102176453792067413801, 2.98289897641336701795063359284, 4.281172027307353707398122002520, 5.15335549885433772475709827335, 6.539699000643124245940270355970, 8.47189333737247937318373128997, 9.64122700639425801436451320974, 10.51004081074034481544353427778, 11.16928808259958798044487728572, 13.157013048649216013163617009316, 13.58393325435369317402000647794, 14.52265785585738182469328963231, 15.64379696370361329411241010327, 16.81873463454544308473054255084, 18.1336250581457355335599686835, 19.33854019047833017185104586496, 20.5627338914319429965569189899, 20.931045628004634315349897839266, 21.70990407464236792512205228282, 22.86638498457824168196432009931, 23.78907101012283397766441805628, 25.05541107261038113934081214579, 26.11169725672861159612746053486, 26.98411047744662541301389576355, 28.14249042863131491786433544856

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