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Properties

Label 1-149-149.133-r0-0-0
Degree $1$
Conductor $149$
Sign $-0.316 - 0.948i$
Analytic cond. $0.691953$
Root an. cond. $0.691953$
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  + (−0.985 − 0.169i)2-s + (0.594 − 0.803i)3-s + (0.942 + 0.333i)4-s + (0.372 − 0.927i)5-s + (−0.721 + 0.691i)6-s + (0.524 − 0.851i)7-s + (−0.873 − 0.487i)8-s + (−0.292 − 0.956i)9-s + (−0.524 + 0.851i)10-s + (−0.942 + 0.333i)11-s + (0.828 − 0.559i)12-s + (0.911 − 0.411i)13-s + (−0.660 + 0.750i)14-s + (−0.524 − 0.851i)15-s + (0.778 + 0.628i)16-s + (−0.594 + 0.803i)17-s + ⋯
L(s)  = 1  + (−0.985 − 0.169i)2-s + (0.594 − 0.803i)3-s + (0.942 + 0.333i)4-s + (0.372 − 0.927i)5-s + (−0.721 + 0.691i)6-s + (0.524 − 0.851i)7-s + (−0.873 − 0.487i)8-s + (−0.292 − 0.956i)9-s + (−0.524 + 0.851i)10-s + (−0.942 + 0.333i)11-s + (0.828 − 0.559i)12-s + (0.911 − 0.411i)13-s + (−0.660 + 0.750i)14-s + (−0.524 − 0.851i)15-s + (0.778 + 0.628i)16-s + (−0.594 + 0.803i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $-0.316 - 0.948i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ -0.316 - 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5368915597 - 0.7447610179i\)
\(L(\frac12)\) \(\approx\) \(0.5368915597 - 0.7447610179i\)
\(L(1)\) \(\approx\) \(0.7500532699 - 0.4846121268i\)
\(L(1)\) \(\approx\) \(0.7500532699 - 0.4846121268i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (-0.985 - 0.169i)T \)
3 \( 1 + (0.594 - 0.803i)T \)
5 \( 1 + (0.372 - 0.927i)T \)
7 \( 1 + (0.524 - 0.851i)T \)
11 \( 1 + (-0.942 + 0.333i)T \)
13 \( 1 + (0.911 - 0.411i)T \)
17 \( 1 + (-0.594 + 0.803i)T \)
19 \( 1 + (-0.127 + 0.991i)T \)
23 \( 1 + (0.911 + 0.411i)T \)
29 \( 1 + (0.0424 + 0.999i)T \)
31 \( 1 + (-0.911 - 0.411i)T \)
37 \( 1 + (0.942 - 0.333i)T \)
41 \( 1 + (-0.778 - 0.628i)T \)
43 \( 1 + (0.996 + 0.0848i)T \)
47 \( 1 + (-0.127 - 0.991i)T \)
53 \( 1 + (-0.996 + 0.0848i)T \)
59 \( 1 + (0.967 + 0.251i)T \)
61 \( 1 + (0.985 + 0.169i)T \)
67 \( 1 + (0.660 + 0.750i)T \)
71 \( 1 + (-0.372 + 0.927i)T \)
73 \( 1 + (-0.450 + 0.892i)T \)
79 \( 1 + (0.450 + 0.892i)T \)
83 \( 1 + (0.450 - 0.892i)T \)
89 \( 1 + (-0.210 - 0.977i)T \)
97 \( 1 + (0.996 - 0.0848i)T \)
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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

−28.32290586170597442610414729831, −27.12547893199612305940075655268, −26.56426445828377093775141968185, −25.65855761501961789648829526744, −25.02780188098050794816164236702, −23.728854549317277177019537296866, −22.19893187157502529326929201579, −21.23418624585176927669305366593, −20.625578441991535530991218946190, −19.19141958188266351386359549697, −18.49098765793864879090261470856, −17.63006296674187760162229721263, −16.150254988834127888554177748244, −15.43618861441859271412247215655, −14.641418812825531224785956570633, −13.44486471324764811277609005260, −11.22700396105683435378950906429, −10.93710223600490421965574618382, −9.567964503844385652639963291644, −8.794919839980238127402183105341, −7.7334371177271939573858724620, −6.34163605999247922401581843696, −5.06010240995217706184175976627, −2.98367264079714346323319732038, −2.225651340134132413477314551197, 1.08429435858670912867765727885, 2.04425851097635880084726098441, 3.73575505721037856526609971777, 5.71138936108833777095738370025, 7.14496007061924614421315765613, 8.118741924827938735665006981717, 8.77812848689339580982650579411, 10.10561129790385632425352405239, 11.19260983776153361770390379002, 12.71182861709811764172866687523, 13.19820401713228593633920482117, 14.68677941377197579019053168063, 15.97309021004867416349208631521, 17.1470810195824795454168845721, 17.850540325356668905281590698903, 18.75928671156781656523737976198, 20.04596473729739370210219082935, 20.51402162226486956132861949911, 21.26005212867497946608709125456, 23.48055830330702917637704473229, 24.03054300324013866985994260646, 25.18731234600942236167514824707, 25.73974064029342528796050730522, 26.79660400071672142172833583137, 27.85418452003462480575834416877

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