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Properties

Label 1-151-151.2-r0-0-0
Degree $1$
Conductor $151$
Sign $0.743 - 0.668i$
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  + (−0.5 − 0.866i)2-s + (0.309 − 0.951i)3-s + (−0.5 + 0.866i)4-s + (−0.104 + 0.994i)5-s + (−0.978 + 0.207i)6-s + (−0.104 + 0.994i)7-s + 8-s + (−0.809 − 0.587i)9-s + (0.913 − 0.406i)10-s + (0.669 − 0.743i)11-s + (0.669 + 0.743i)12-s + (0.669 + 0.743i)13-s + (0.913 − 0.406i)14-s + (0.913 + 0.406i)15-s + (−0.5 − 0.866i)16-s + (0.913 + 0.406i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.309 − 0.951i)3-s + (−0.5 + 0.866i)4-s + (−0.104 + 0.994i)5-s + (−0.978 + 0.207i)6-s + (−0.104 + 0.994i)7-s + 8-s + (−0.809 − 0.587i)9-s + (0.913 − 0.406i)10-s + (0.669 − 0.743i)11-s + (0.669 + 0.743i)12-s + (0.669 + 0.743i)13-s + (0.913 − 0.406i)14-s + (0.913 + 0.406i)15-s + (−0.5 − 0.866i)16-s + (0.913 + 0.406i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.743 - 0.668i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.743 - 0.668i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8593348588 - 0.3295503329i\)
\(L(\frac12)\) \(\approx\) \(0.8593348588 - 0.3295503329i\)
\(L(1)\) \(\approx\) \(0.8433620725 - 0.3147232855i\)
\(L(1)\) \(\approx\) \(0.8433620725 - 0.3147232855i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-0.104 + 0.994i)T \)
7 \( 1 + (-0.104 + 0.994i)T \)
11 \( 1 + (0.669 - 0.743i)T \)
13 \( 1 + (0.669 + 0.743i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.104 - 0.994i)T \)
47 \( 1 + (0.669 + 0.743i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.978 - 0.207i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (0.913 - 0.406i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (-0.104 - 0.994i)T \)
97 \( 1 + (0.913 + 0.406i)T \)
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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

−27.841955976590114704241153746212, −27.24806540845600004656456605223, −26.21748362317627854441540813345, −25.350940881818215153266194062827, −24.55585178842955105775521765041, −23.17618264943553877817810269990, −22.70863848678475325749942081420, −20.999767375053839112983675522846, −20.15502883852456720857345690260, −19.5275178542049255015220071930, −17.758337092606421883832549297434, −16.97124083011650441135098387119, −16.15038363395729097512004571762, −15.41408976669553296172187032500, −14.1599928387314476110738182903, −13.37665649225289308630716501292, −11.60048355081007875499236946354, −10.037706160451946381719064687040, −9.62637100427075804497226528090, −8.33617073990312736157921017625, −7.49940257874124427244940893498, −5.794325033625036237162614644109, −4.72851201004555808735361380665, −3.73949567402603112196674529951, −1.127896010991184601269731984784, 1.45345088416927883447074536285, 2.74199719132087899753279774493, 3.59192075357945664528333853054, 5.93296561156918358236269784179, 7.06294941008043645708765320232, 8.34124422774492695587146681622, 9.111926573569397902965718120468, 10.56710163297990501465824439555, 11.78724097944661129553714590404, 12.20394942218164763757584746322, 13.77292826528083890815636093984, 14.35476189419953043910878151711, 16.02131612719442047887601627225, 17.41400826231383531380233791740, 18.58732870381524440530039487666, 18.74914308137101238828250982852, 19.66185204615503694663462752411, 20.96711796889659250618865554431, 21.997399078068592584283926163737, 22.78552567906813633303487565893, 24.09603021721073638572527768578, 25.32946725652346550057959007872, 26.006388143676822173500371956538, 26.954492252570689895113059014501, 28.15992431041509804674873848977

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