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Properties

Label 1-151-151.74-r0-0-0
Degree $1$
Conductor $151$
Sign $-0.999 + 0.00227i$
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 + (−0.929 + 0.368i)3-s + (−0.104 + 0.994i)4-s + (0.228 + 0.973i)5-s + (−0.895 − 0.444i)6-s + (−0.855 + 0.518i)7-s + (−0.809 + 0.587i)8-s + (0.728 − 0.684i)9-s + (−0.570 + 0.821i)10-s + (0.783 − 0.621i)11-s + (−0.268 − 0.963i)12-s + (−0.999 − 0.0418i)13-s + (−0.957 − 0.289i)14-s + (−0.570 − 0.821i)15-s + (−0.978 − 0.207i)16-s + (−0.0209 + 0.999i)17-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (−0.929 + 0.368i)3-s + (−0.104 + 0.994i)4-s + (0.228 + 0.973i)5-s + (−0.895 − 0.444i)6-s + (−0.855 + 0.518i)7-s + (−0.809 + 0.587i)8-s + (0.728 − 0.684i)9-s + (−0.570 + 0.821i)10-s + (0.783 − 0.621i)11-s + (−0.268 − 0.963i)12-s + (−0.999 − 0.0418i)13-s + (−0.957 − 0.289i)14-s + (−0.570 − 0.821i)15-s + (−0.978 − 0.207i)16-s + (−0.0209 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $-0.999 + 0.00227i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ -0.999 + 0.00227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001027284643 + 0.9017513577i\)
\(L(\frac12)\) \(\approx\) \(0.001027284643 + 0.9017513577i\)
\(L(1)\) \(\approx\) \(0.6041500947 + 0.7480270953i\)
\(L(1)\) \(\approx\) \(0.6041500947 + 0.7480270953i\)

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 \( 1 + (-0.929 + 0.368i)T \)
5 \( 1 + (0.228 + 0.973i)T \)
7 \( 1 + (-0.855 + 0.518i)T \)
11 \( 1 + (0.783 - 0.621i)T \)
13 \( 1 + (-0.999 - 0.0418i)T \)
17 \( 1 + (-0.0209 + 0.999i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.913 - 0.406i)T \)
29 \( 1 + (-0.637 + 0.770i)T \)
31 \( 1 + (0.944 + 0.328i)T \)
37 \( 1 + (0.463 + 0.886i)T \)
41 \( 1 + (0.0627 + 0.998i)T \)
43 \( 1 + (-0.855 - 0.518i)T \)
47 \( 1 + (0.832 - 0.553i)T \)
53 \( 1 + (0.968 - 0.248i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.146 + 0.989i)T \)
67 \( 1 + (0.728 + 0.684i)T \)
71 \( 1 + (-0.0209 - 0.999i)T \)
73 \( 1 + (0.876 + 0.481i)T \)
79 \( 1 + (-0.187 + 0.982i)T \)
83 \( 1 + (-0.992 - 0.125i)T \)
89 \( 1 + (0.387 + 0.921i)T \)
97 \( 1 + (-0.957 + 0.289i)T \)
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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

−27.919139473338526304309722939131, −27.088558449226191787156735818123, −25.083888023283230192737085522, −24.52806170406313387862332683641, −23.28333076127415177061890133197, −22.78265495379109802308739129642, −21.8011488114409957143192056141, −20.700265143407479950380049050396, −19.701329683262127319149102563322, −18.92136364550849437289782725875, −17.40041270043748691595695558203, −16.741853905177320667230657787029, −15.502905319170730458026375355524, −13.99040871740033501005968687236, −12.9478178810820943576247951340, −12.37936807141526779022351905737, −11.453091913062916496128908491, −10.04721459824612462389078308237, −9.388297374454951973416120149185, −7.22549627380901796414436489378, −6.13298323516225258939633714433, −4.97966200025846536088354275752, −4.08536986055277827758932138307, −2.1308793917939336055184404990, −0.69538322375362203339309404605, 2.80562597209230735805687733189, 3.9904763813524169091780303519, 5.402619681706238882680043772724, 6.43641537358000254939635791953, 6.90416893958669801320087765115, 8.798080510665979343916978022321, 10.07538376500905149285141002829, 11.30956020711332142445937477697, 12.31260615110950119264363725471, 13.31670090371838879615647130576, 14.84685422921784532616422331360, 15.22700376615982853177361026954, 16.64463430250116178511890735626, 17.155948494534493483948086052184, 18.37199986885825907449100321672, 19.4644356089507756034545883532, 21.45307924585944266670797348706, 21.92620279283521136268829728308, 22.54184390754454985959285391053, 23.468919140270681885575298018, 24.54353823634414261393747066383, 25.59098404043940562868539352799, 26.54186633786817438332219914785, 27.28925083038127853114458435872, 28.693306982521830460532027258067

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