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Properties

Label 1-151-151.121-r0-0-0
Degree $1$
Conductor $151$
Sign $0.116 - 0.993i$
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  + (−0.104 − 0.994i)2-s + (−0.992 − 0.125i)3-s + (−0.978 + 0.207i)4-s + (−0.699 + 0.714i)5-s + (−0.0209 + 0.999i)6-s + (0.985 − 0.166i)7-s + (0.309 + 0.951i)8-s + (0.968 + 0.248i)9-s + (0.783 + 0.621i)10-s + (0.387 − 0.921i)11-s + (0.996 − 0.0836i)12-s + (−0.756 + 0.653i)13-s + (−0.268 − 0.963i)14-s + (0.783 − 0.621i)15-s + (0.913 − 0.406i)16-s + (−0.348 − 0.937i)17-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.992 − 0.125i)3-s + (−0.978 + 0.207i)4-s + (−0.699 + 0.714i)5-s + (−0.0209 + 0.999i)6-s + (0.985 − 0.166i)7-s + (0.309 + 0.951i)8-s + (0.968 + 0.248i)9-s + (0.783 + 0.621i)10-s + (0.387 − 0.921i)11-s + (0.996 − 0.0836i)12-s + (−0.756 + 0.653i)13-s + (−0.268 − 0.963i)14-s + (0.783 − 0.621i)15-s + (0.913 − 0.406i)16-s + (−0.348 − 0.937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.116 - 0.993i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.116 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4948681126 - 0.4400084435i\)
\(L(\frac12)\) \(\approx\) \(0.4948681126 - 0.4400084435i\)
\(L(1)\) \(\approx\) \(0.6177165397 - 0.3211758259i\)
\(L(1)\) \(\approx\) \(0.6177165397 - 0.3211758259i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (-0.104 - 0.994i)T \)
3 \( 1 + (-0.992 - 0.125i)T \)
5 \( 1 + (-0.699 + 0.714i)T \)
7 \( 1 + (0.985 - 0.166i)T \)
11 \( 1 + (0.387 - 0.921i)T \)
13 \( 1 + (-0.756 + 0.653i)T \)
17 \( 1 + (-0.348 - 0.937i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.669 + 0.743i)T \)
29 \( 1 + (0.728 - 0.684i)T \)
31 \( 1 + (0.832 + 0.553i)T \)
37 \( 1 + (0.944 + 0.328i)T \)
41 \( 1 + (0.876 - 0.481i)T \)
43 \( 1 + (0.985 + 0.166i)T \)
47 \( 1 + (-0.855 - 0.518i)T \)
53 \( 1 + (-0.425 - 0.904i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (0.604 + 0.796i)T \)
67 \( 1 + (0.968 - 0.248i)T \)
71 \( 1 + (-0.348 + 0.937i)T \)
73 \( 1 + (-0.637 - 0.770i)T \)
79 \( 1 + (0.0627 + 0.998i)T \)
83 \( 1 + (0.535 + 0.844i)T \)
89 \( 1 + (0.463 - 0.886i)T \)
97 \( 1 + (-0.268 + 0.963i)T \)
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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

−27.891040817170868701479871093, −27.46380676076584689939104896099, −26.60460567609006949651520812073, −24.86468758678461016692111671511, −24.48706124908878914454625155700, −23.42164327223428269309597377806, −22.78816024703673141969113815820, −21.691480801291053483710272601885, −20.432817913199318482940149501609, −19.0856891370664729094449910514, −17.8118951514857347118951762073, −17.276246153623096993799025372756, −16.362967258304219800870560647423, −15.281330011708533936029448100546, −14.61928791989378243461149502373, −12.763135295992729366721027986486, −12.19766501734219865218024939379, −10.7763707828829175602335593631, −9.53597173873860399058113086030, −8.19206584097550767901345498137, −7.36592750099682278088274024731, −5.987188224234213090941033893772, −4.817010053772291372717098724757, −4.303803156284202334030469158078, −1.18467519657882286069422479352, 0.8861441019849143824140143010, 2.62991291353890251647504485248, 4.20922352618572109614242423400, 5.0885673125327580228736802341, 6.82913351655676417361437312532, 7.9510013631056257583134056893, 9.442527195380864648133749542334, 10.79913218765650529439992545378, 11.47621055225763649050612276978, 11.88646318675328711287285938920, 13.48874066800702534645320896213, 14.44647463027204880981651663142, 15.90212161366223972052885412009, 17.23555163207524081740669049563, 17.93373282145027762773580277854, 18.96106842346665494529562951327, 19.67264044666617778990207977695, 21.18881232354254264486461513029, 21.875792051469627780693938822580, 22.769454529970164804127635041116, 23.66935535623001021542971186882, 24.51682647518189323205147679943, 26.6253973900811048611847565707, 27.04408602040543595905712558411, 27.78724428380617254466407262654

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