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Properties

Label 1-151-151.127-r0-0-0
Degree $1$
Conductor $151$
Sign $0.393 + 0.919i$
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  + (0.309 − 0.951i)2-s + (−0.637 − 0.770i)3-s + (−0.809 − 0.587i)4-s + (−0.187 + 0.982i)5-s + (−0.929 + 0.368i)6-s + (−0.992 + 0.125i)7-s + (−0.809 + 0.587i)8-s + (−0.187 + 0.982i)9-s + (0.876 + 0.481i)10-s + (−0.637 + 0.770i)11-s + (0.0627 + 0.998i)12-s + (0.968 + 0.248i)13-s + (−0.187 + 0.982i)14-s + (0.876 − 0.481i)15-s + (0.309 + 0.951i)16-s + (−0.992 − 0.125i)17-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.637 − 0.770i)3-s + (−0.809 − 0.587i)4-s + (−0.187 + 0.982i)5-s + (−0.929 + 0.368i)6-s + (−0.992 + 0.125i)7-s + (−0.809 + 0.587i)8-s + (−0.187 + 0.982i)9-s + (0.876 + 0.481i)10-s + (−0.637 + 0.770i)11-s + (0.0627 + 0.998i)12-s + (0.968 + 0.248i)13-s + (−0.187 + 0.982i)14-s + (0.876 − 0.481i)15-s + (0.309 + 0.951i)16-s + (−0.992 − 0.125i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.393 + 0.919i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.393 + 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2062960213 + 0.1361048219i\)
\(L(\frac12)\) \(\approx\) \(0.2062960213 + 0.1361048219i\)
\(L(1)\) \(\approx\) \(0.5389431542 - 0.2189846596i\)
\(L(1)\) \(\approx\) \(0.5389431542 - 0.2189846596i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (-0.637 - 0.770i)T \)
5 \( 1 + (-0.187 + 0.982i)T \)
7 \( 1 + (-0.992 + 0.125i)T \)
11 \( 1 + (-0.637 + 0.770i)T \)
13 \( 1 + (0.968 + 0.248i)T \)
17 \( 1 + (-0.992 - 0.125i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (0.535 + 0.844i)T \)
31 \( 1 + (-0.425 + 0.904i)T \)
37 \( 1 + (0.968 + 0.248i)T \)
41 \( 1 + (-0.929 + 0.368i)T \)
43 \( 1 + (-0.992 - 0.125i)T \)
47 \( 1 + (-0.929 + 0.368i)T \)
53 \( 1 + (0.0627 - 0.998i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (-0.637 + 0.770i)T \)
67 \( 1 + (-0.187 - 0.982i)T \)
71 \( 1 + (-0.992 + 0.125i)T \)
73 \( 1 + (-0.992 + 0.125i)T \)
79 \( 1 + (-0.425 - 0.904i)T \)
83 \( 1 + (0.728 + 0.684i)T \)
89 \( 1 + (0.728 + 0.684i)T \)
97 \( 1 + (-0.187 - 0.982i)T \)
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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

−27.81765092587024969525520580374, −26.7700733304491234914638842412, −25.98334153452618380090622582256, −24.92810336296686871103952528288, −23.64276479464284686020615468655, −23.2903607484338049788494239286, −22.08621268328953051283854860354, −21.29516389065925494758852541077, −20.203470699832994173708108217037, −18.67121163737317915901254741730, −17.44590467325479353555211077045, −16.51026240455844043327932946493, −15.96323061210864670480988641290, −15.26063512768664767465899342543, −13.50820699540622959874829016320, −12.8894450857867253724049823818, −11.60804477241171624496253154035, −10.13327825869176702987922923275, −9.018352735144680902620864682070, −8.11538496430002460101892993659, −6.30549803245333009556830131220, −5.71752986180069222054881002393, −4.38551399000034399071513345210, −3.53368916394659393196067342696, −0.19843965744099275320005258527, 1.98647520413745668859605922538, 3.05683740927509078619130883993, 4.5824290253923256438667328736, 6.1496166804941927825842717299, 6.86454028226478408938002500752, 8.56385090447500634353291221138, 10.15366917629948304564944746166, 10.86560571060366855555663364295, 11.88330224568903725458409758776, 12.94198071279195981684975153576, 13.58030489150934568873338265332, 14.96490808861801468341888979032, 16.1720554253185507312612672584, 17.95730762092174822796144058117, 18.25406536226746453115343182485, 19.326902022911090263072638530144, 20.063831004790145688035791509961, 21.67797202105665776820940915500, 22.37393391906092748482956117100, 23.22833447802335540804980038954, 23.73932009648645697333508626975, 25.4491885597415423116004051661, 26.29007213921359020960224226663, 27.62852227120084082352427441027, 28.70137320490576780034940787758

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