↑

Properties

Label 1-149-149.112-r0-0-0
Degree $1$
Conductor $149$
Sign $0.447 - 0.894i$
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  + (0.996 − 0.0848i)2-s + (0.450 − 0.892i)3-s + (0.985 − 0.169i)4-s + (−0.828 − 0.559i)5-s + (0.372 − 0.927i)6-s + (0.873 + 0.487i)7-s + (0.967 − 0.251i)8-s + (−0.594 − 0.803i)9-s + (−0.873 − 0.487i)10-s + (−0.985 − 0.169i)11-s + (0.292 − 0.956i)12-s + (−0.210 + 0.977i)13-s + (0.911 + 0.411i)14-s + (−0.873 + 0.487i)15-s + (0.942 − 0.333i)16-s + (−0.450 + 0.892i)17-s + ⋯
L(s)  = 1  + (0.996 − 0.0848i)2-s + (0.450 − 0.892i)3-s + (0.985 − 0.169i)4-s + (−0.828 − 0.559i)5-s + (0.372 − 0.927i)6-s + (0.873 + 0.487i)7-s + (0.967 − 0.251i)8-s + (−0.594 − 0.803i)9-s + (−0.873 − 0.487i)10-s + (−0.985 − 0.169i)11-s + (0.292 − 0.956i)12-s + (−0.210 + 0.977i)13-s + (0.911 + 0.411i)14-s + (−0.873 + 0.487i)15-s + (0.942 − 0.333i)16-s + (−0.450 + 0.892i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.748973193 - 1.080935122i\)
\(L(\frac12)\) \(\approx\) \(1.748973193 - 1.080935122i\)
\(L(1)\) \(\approx\) \(1.734044600 - 0.6792166648i\)
\(L(1)\) \(\approx\) \(1.734044600 - 0.6792166648i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (0.996 - 0.0848i)T \)
3 \( 1 + (0.450 - 0.892i)T \)
5 \( 1 + (-0.828 - 0.559i)T \)
7 \( 1 + (0.873 + 0.487i)T \)
11 \( 1 + (-0.985 - 0.169i)T \)
13 \( 1 + (-0.210 + 0.977i)T \)
17 \( 1 + (-0.450 + 0.892i)T \)
19 \( 1 + (0.660 - 0.750i)T \)
23 \( 1 + (-0.210 - 0.977i)T \)
29 \( 1 + (-0.721 + 0.691i)T \)
31 \( 1 + (0.210 + 0.977i)T \)
37 \( 1 + (0.985 + 0.169i)T \)
41 \( 1 + (-0.942 + 0.333i)T \)
43 \( 1 + (-0.0424 - 0.999i)T \)
47 \( 1 + (0.660 + 0.750i)T \)
53 \( 1 + (0.0424 - 0.999i)T \)
59 \( 1 + (0.127 + 0.991i)T \)
61 \( 1 + (-0.996 + 0.0848i)T \)
67 \( 1 + (-0.911 + 0.411i)T \)
71 \( 1 + (0.828 + 0.559i)T \)
73 \( 1 + (0.524 - 0.851i)T \)
79 \( 1 + (-0.524 - 0.851i)T \)
83 \( 1 + (-0.524 + 0.851i)T \)
89 \( 1 + (-0.778 + 0.628i)T \)
97 \( 1 + (-0.0424 + 0.999i)T \)
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

−28.16951629505106238050269610859, −27.09857054724767690320268497575, −26.44192645072438195809986936773, −25.30035846267219461425069417582, −24.234538814939128133135828541672, −23.12023256071674448435445192032, −22.52545243640039727699743656473, −21.38931198076435363109312408752, −20.399762014575800925740705243769, −20.00488041877757329096408013051, −18.39850236091443920949866697601, −16.90092510225986786900186723856, −15.64923575367235155549434545157, −15.25473171127623023948814651777, −14.24064082070657472235143250149, −13.3247821360858643394683425193, −11.71156055954714780514373465798, −10.9692701914203671732566682458, −10.01236266134558252835827927627, −7.87339214113027674804722868615, −7.60542898804408870991217177465, −5.554267852908533400744326562318, −4.582110212607980522907046661017, −3.54981190627724307553445082467, −2.487319519001913705056528786883, 1.59613996875612152322295629407, 2.80826370343577888999048883869, 4.30375428295804226303673549596, 5.39127467596214495017372884072, 6.8575410444575457638803235534, 7.88234332829363861147696006254, 8.80160070091308223589154490880, 10.95130377633904787564020383349, 11.86525087994035641746583187301, 12.630426303900695538109501715387, 13.611537219099963446225151320646, 14.69210810810088086266889827680, 15.492161280453410691407836607262, 16.701105709245878467955822714584, 18.19907830423476218184044684335, 19.21499630967702077488707432183, 20.15480792691051029157650579495, 20.930492582803247792165824277924, 22.008757444700604217847829100630, 23.490970474907391490552410222397, 24.04070653427965396209690544634, 24.4482313993101398764351518560, 25.73164208712972915797296672861, 26.81357908741252054650514651569, 28.609122887593641964844033477139

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