↑

Properties

Label 1-149-149.88-r0-0-0
Degree $1$
Conductor $149$
Sign $-0.910 + 0.414i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0424 − 0.999i)2-s + (0.524 − 0.851i)3-s + (−0.996 − 0.0848i)4-s + (−0.292 − 0.956i)5-s + (−0.828 − 0.559i)6-s + (−0.967 + 0.251i)7-s + (−0.127 + 0.991i)8-s + (−0.450 − 0.892i)9-s + (−0.967 + 0.251i)10-s + (−0.996 + 0.0848i)11-s + (−0.594 + 0.803i)12-s + (0.778 + 0.628i)13-s + (0.210 + 0.977i)14-s + (−0.967 − 0.251i)15-s + (0.985 + 0.169i)16-s + (0.524 − 0.851i)17-s + ⋯
L(s)  = 1  + (0.0424 − 0.999i)2-s + (0.524 − 0.851i)3-s + (−0.996 − 0.0848i)4-s + (−0.292 − 0.956i)5-s + (−0.828 − 0.559i)6-s + (−0.967 + 0.251i)7-s + (−0.127 + 0.991i)8-s + (−0.450 − 0.892i)9-s + (−0.967 + 0.251i)10-s + (−0.996 + 0.0848i)11-s + (−0.594 + 0.803i)12-s + (0.778 + 0.628i)13-s + (0.210 + 0.977i)14-s + (−0.967 − 0.251i)15-s + (0.985 + 0.169i)16-s + (0.524 − 0.851i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $-0.910 + 0.414i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ -0.910 + 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1705020047 - 0.7855772874i\)
\(L(\frac12)\) \(\approx\) \(-0.1705020047 - 0.7855772874i\)
\(L(1)\) \(\approx\) \(0.4533348865 - 0.7558047101i\)
\(L(1)\) \(\approx\) \(0.4533348865 - 0.7558047101i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (0.0424 - 0.999i)T \)
3 \( 1 + (0.524 - 0.851i)T \)
5 \( 1 + (-0.292 - 0.956i)T \)
7 \( 1 + (-0.967 + 0.251i)T \)
11 \( 1 + (-0.996 + 0.0848i)T \)
13 \( 1 + (0.778 + 0.628i)T \)
17 \( 1 + (0.524 - 0.851i)T \)
19 \( 1 + (-0.911 - 0.411i)T \)
23 \( 1 + (0.778 - 0.628i)T \)
29 \( 1 + (0.372 - 0.927i)T \)
31 \( 1 + (0.778 - 0.628i)T \)
37 \( 1 + (-0.996 + 0.0848i)T \)
41 \( 1 + (0.985 + 0.169i)T \)
43 \( 1 + (-0.721 + 0.691i)T \)
47 \( 1 + (-0.911 + 0.411i)T \)
53 \( 1 + (-0.721 - 0.691i)T \)
59 \( 1 + (0.660 + 0.750i)T \)
61 \( 1 + (0.0424 - 0.999i)T \)
67 \( 1 + (0.210 - 0.977i)T \)
71 \( 1 + (-0.292 - 0.956i)T \)
73 \( 1 + (0.873 + 0.487i)T \)
79 \( 1 + (0.873 - 0.487i)T \)
83 \( 1 + (0.873 + 0.487i)T \)
89 \( 1 + (0.942 + 0.333i)T \)
97 \( 1 + (-0.721 - 0.691i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.274072709743886389696529114876, −27.33435647899550050474938604869, −26.442084775731229371085256063829, −25.80448577222346039579801406996, −25.28660464953478068977027669325, −23.41243711246939243370680319965, −23.031362328829281996629675469606, −21.944172838252209356368962850030, −21.042233338080419562920665790964, −19.452226436848988021675549675047, −18.80802870658495378690068524396, −17.48511166921776214650805618248, −16.23200618958439201019413992114, −15.58252825263604000733096827708, −14.82002166972581948151428184804, −13.74529356345938872886765800925, −12.794554005067825263946973970253, −10.60870566100216626842487390329, −10.15953302481477706392269077107, −8.69696877473890099338028851109, −7.78097084304546595932141832812, −6.530660272695524112236827123203, −5.37369529381794774123016227767, −3.79124264284500218536520623817, −3.102367283078041769581446948005, 0.64678084373759374048320614053, 2.238053258387227400917303997309, 3.35395812125817497830455743015, 4.78593700945961947471781587848, 6.29331587191161791470534552428, 7.97542141737782851122752631964, 8.85994135398742499446718017874, 9.75349212008009121618191775467, 11.36395590509726489664207964282, 12.426699460748963348571877377150, 13.06273199619960644530317322233, 13.78056347540472563618449086264, 15.352983984407176588997726448391, 16.632623282055179603054261890271, 17.9551609075846619685897674067, 19.01409310490212934398409594868, 19.415793591393640097655626238956, 20.748947138102096609165593146307, 21.05909309913276565488161890626, 22.90077624438693145902703510655, 23.40282748360589285545989304687, 24.53765885863986260482259837719, 25.72458443784446485200007323086, 26.52388387069311062412064654668, 27.998490036988883150169645503030

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