↑

Properties

Label 1-149-149.100-r0-0-0
Degree $1$
Conductor $149$
Sign $0.542 + 0.840i$
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  + (−0.873 + 0.487i)2-s + (−0.942 + 0.333i)3-s + (0.524 − 0.851i)4-s + (−0.911 − 0.411i)5-s + (0.660 − 0.750i)6-s + (−0.996 + 0.0848i)7-s + (−0.0424 + 0.999i)8-s + (0.778 − 0.628i)9-s + (0.996 − 0.0848i)10-s + (−0.524 − 0.851i)11-s + (−0.210 + 0.977i)12-s + (0.292 + 0.956i)13-s + (0.828 − 0.559i)14-s + (0.996 + 0.0848i)15-s + (−0.450 − 0.892i)16-s + (0.942 − 0.333i)17-s + ⋯
L(s)  = 1  + (−0.873 + 0.487i)2-s + (−0.942 + 0.333i)3-s + (0.524 − 0.851i)4-s + (−0.911 − 0.411i)5-s + (0.660 − 0.750i)6-s + (−0.996 + 0.0848i)7-s + (−0.0424 + 0.999i)8-s + (0.778 − 0.628i)9-s + (0.996 − 0.0848i)10-s + (−0.524 − 0.851i)11-s + (−0.210 + 0.977i)12-s + (0.292 + 0.956i)13-s + (0.828 − 0.559i)14-s + (0.996 + 0.0848i)15-s + (−0.450 − 0.892i)16-s + (0.942 − 0.333i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.542 + 0.840i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.542 + 0.840i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3166775435 + 0.1725030125i\)
\(L(\frac12)\) \(\approx\) \(0.3166775435 + 0.1725030125i\)
\(L(1)\) \(\approx\) \(0.4204240790 + 0.1110062257i\)
\(L(1)\) \(\approx\) \(0.4204240790 + 0.1110062257i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (-0.873 + 0.487i)T \)
3 \( 1 + (-0.942 + 0.333i)T \)
5 \( 1 + (-0.911 - 0.411i)T \)
7 \( 1 + (-0.996 + 0.0848i)T \)
11 \( 1 + (-0.524 - 0.851i)T \)
13 \( 1 + (0.292 + 0.956i)T \)
17 \( 1 + (0.942 - 0.333i)T \)
19 \( 1 + (0.372 + 0.927i)T \)
23 \( 1 + (0.292 - 0.956i)T \)
29 \( 1 + (-0.127 + 0.991i)T \)
31 \( 1 + (-0.292 + 0.956i)T \)
37 \( 1 + (0.524 + 0.851i)T \)
41 \( 1 + (0.450 + 0.892i)T \)
43 \( 1 + (0.967 - 0.251i)T \)
47 \( 1 + (0.372 - 0.927i)T \)
53 \( 1 + (-0.967 - 0.251i)T \)
59 \( 1 + (0.721 - 0.691i)T \)
61 \( 1 + (0.873 - 0.487i)T \)
67 \( 1 + (-0.828 - 0.559i)T \)
71 \( 1 + (0.911 + 0.411i)T \)
73 \( 1 + (0.985 + 0.169i)T \)
79 \( 1 + (-0.985 + 0.169i)T \)
83 \( 1 + (-0.985 - 0.169i)T \)
89 \( 1 + (0.594 - 0.803i)T \)
97 \( 1 + (0.967 + 0.251i)T \)
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

−28.05864616182301098834227160620, −27.28120707832802630002842610721, −26.11107546017473472867021804290, −25.34420047602974944166140114334, −23.88959242443801434125889613454, −22.86554644758610929145732615377, −22.294837119967197312492249713705, −20.86931316866166354150744712540, −19.65575540350985429803843924741, −19.01273550114625351057909866695, −18.036234635765930522104450897778, −17.177235249458181121184640573636, −15.95879550588663826104148342826, −15.44168396656209045068212908249, −13.08222416364003701028649218648, −12.44863280884050161339758454584, −11.3798350058207498650791164221, −10.50299203329124863979586409508, −9.56686442527431022368797760356, −7.720968281541209439736649917479, −7.26718179207492830347093072643, −5.866739237508679475221683380670, −3.99524031645493770912993385629, −2.67892930477535748246083991263, −0.66481908163156406134696210350, 0.92612832887872946192378192218, 3.433775975660364019048717141375, 5.04783303242791447123289352572, 6.15568925153182312096706969350, 7.17974043253363283307848898982, 8.497754592801782078197585102520, 9.594892385690285445057997737, 10.667641471916564735715346429481, 11.6462127514868471043227116970, 12.649734694895057456879701356353, 14.426245283693183687993281722107, 15.86879053438176026791320075776, 16.274659970668663561135873141899, 16.819440416505143339684884351119, 18.59573379947716622212553507910, 18.80766438974226646037705228109, 20.1901769396627707135473237209, 21.28756529217184895474546240465, 22.74322020933435395669534475011, 23.4942796164312392402904478534, 24.201476769966905798488303571750, 25.46980723613883247854056246977, 26.703961788025549471663362926, 27.136291320867543293168745153179, 28.31636281929226677873616921041

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