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Properties

Label 1-151-151.123-r0-0-0
Degree $1$
Conductor $151$
Sign $0.975 + 0.222i$
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  + (−0.809 − 0.587i)2-s + (−0.187 + 0.982i)3-s + (0.309 + 0.951i)4-s + (−0.929 − 0.368i)5-s + (0.728 − 0.684i)6-s + (0.968 − 0.248i)7-s + (0.309 − 0.951i)8-s + (−0.929 − 0.368i)9-s + (0.535 + 0.844i)10-s + (−0.187 − 0.982i)11-s + (−0.992 + 0.125i)12-s + (0.876 + 0.481i)13-s + (−0.929 − 0.368i)14-s + (0.535 − 0.844i)15-s + (−0.809 + 0.587i)16-s + (0.968 + 0.248i)17-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.187 + 0.982i)3-s + (0.309 + 0.951i)4-s + (−0.929 − 0.368i)5-s + (0.728 − 0.684i)6-s + (0.968 − 0.248i)7-s + (0.309 − 0.951i)8-s + (−0.929 − 0.368i)9-s + (0.535 + 0.844i)10-s + (−0.187 − 0.982i)11-s + (−0.992 + 0.125i)12-s + (0.876 + 0.481i)13-s + (−0.929 − 0.368i)14-s + (0.535 − 0.844i)15-s + (−0.809 + 0.587i)16-s + (0.968 + 0.248i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.975 + 0.222i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.975 + 0.222i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6662314095 + 0.07492928887i\)
\(L(\frac12)\) \(\approx\) \(0.6662314095 + 0.07492928887i\)
\(L(1)\) \(\approx\) \(0.6851782973 + 0.01774823599i\)
\(L(1)\) \(\approx\) \(0.6851782973 + 0.01774823599i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (-0.187 + 0.982i)T \)
5 \( 1 + (-0.929 - 0.368i)T \)
7 \( 1 + (0.968 - 0.248i)T \)
11 \( 1 + (-0.187 - 0.982i)T \)
13 \( 1 + (0.876 + 0.481i)T \)
17 \( 1 + (0.968 + 0.248i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (-0.425 + 0.904i)T \)
31 \( 1 + (-0.637 - 0.770i)T \)
37 \( 1 + (0.876 + 0.481i)T \)
41 \( 1 + (0.728 - 0.684i)T \)
43 \( 1 + (0.968 + 0.248i)T \)
47 \( 1 + (0.728 - 0.684i)T \)
53 \( 1 + (-0.992 - 0.125i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (-0.187 - 0.982i)T \)
67 \( 1 + (-0.929 + 0.368i)T \)
71 \( 1 + (0.968 - 0.248i)T \)
73 \( 1 + (0.968 - 0.248i)T \)
79 \( 1 + (-0.637 + 0.770i)T \)
83 \( 1 + (0.0627 + 0.998i)T \)
89 \( 1 + (0.0627 + 0.998i)T \)
97 \( 1 + (-0.929 + 0.368i)T \)
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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

−28.04757282952909501009862988904, −27.12326888126685428244168782743, −25.95037945116079243056874669526, −25.11244003734101356001853691085, −24.17029002429288565336619972307, −23.37229385928135460450073962791, −22.73327072102056570372282568353, −20.66334647150440686221724661249, −19.84578726162472282684273084172, −18.69850100686306078682269805479, −18.1726469942933136523060165522, −17.321762216431839132616485796342, −16.01159567131073046579227555714, −14.98360455344237942607142622759, −14.17675927475175964530514165785, −12.57328093048724547752737845468, −11.43172204841301523184689642805, −10.72291562181650976790819418268, −8.9549121539177567771076803752, −7.80256731345008350306592899332, −7.429678496415729834791535253862, −6.091912885040334695315727398928, −4.80667824165818384814545670434, −2.54740986770874277361696557356, −1.03004000791516544959427624637, 1.18428572809159518535954617387, 3.38179266961204616877504589068, 4.091023129158483513509587493644, 5.61463666730587429269229911827, 7.66820293422985354986191411155, 8.42866121745876543532005708769, 9.42464260980011550237810565510, 10.932318154931076886846772424851, 11.20872559047300295399673310343, 12.31345241019964797644046015144, 13.98614107900512091418949188611, 15.31889251595949299557896737600, 16.41522455628681121947309721860, 16.81472607046274094124548341940, 18.252665909641683839842012619589, 19.193172002017237093315763879975, 20.4240704372441223002281882778, 20.93269613795411747521667862604, 21.78878242917678026276107513317, 23.172644126386081701495969431046, 24.091858056230531032118755765138, 25.569822510898469405287754502097, 26.566219297081268714601753916093, 27.46737763433503510938699114804, 27.66010679436649563852161692306

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