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Properties

Label 1-149-149.39-r0-0-0
Degree $1$
Conductor $149$
Sign $0.336 + 0.941i$
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  + (0.942 − 0.333i)2-s + (−0.292 + 0.956i)3-s + (0.778 − 0.628i)4-s + (−0.721 + 0.691i)5-s + (0.0424 + 0.999i)6-s + (−0.450 + 0.892i)7-s + (0.524 − 0.851i)8-s + (−0.828 − 0.559i)9-s + (−0.450 + 0.892i)10-s + (0.778 + 0.628i)11-s + (0.372 + 0.927i)12-s + (0.660 + 0.750i)13-s + (−0.127 + 0.991i)14-s + (−0.450 − 0.892i)15-s + (0.210 − 0.977i)16-s + (−0.292 + 0.956i)17-s + ⋯
L(s)  = 1  + (0.942 − 0.333i)2-s + (−0.292 + 0.956i)3-s + (0.778 − 0.628i)4-s + (−0.721 + 0.691i)5-s + (0.0424 + 0.999i)6-s + (−0.450 + 0.892i)7-s + (0.524 − 0.851i)8-s + (−0.828 − 0.559i)9-s + (−0.450 + 0.892i)10-s + (0.778 + 0.628i)11-s + (0.372 + 0.927i)12-s + (0.660 + 0.750i)13-s + (−0.127 + 0.991i)14-s + (−0.450 − 0.892i)15-s + (0.210 − 0.977i)16-s + (−0.292 + 0.956i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.336 + 0.941i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.336 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.208323042 + 0.8515182508i\)
\(L(\frac12)\) \(\approx\) \(1.208323042 + 0.8515182508i\)
\(L(1)\) \(\approx\) \(1.337571224 + 0.4443245351i\)
\(L(1)\) \(\approx\) \(1.337571224 + 0.4443245351i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (0.942 - 0.333i)T \)
3 \( 1 + (-0.292 + 0.956i)T \)
5 \( 1 + (-0.721 + 0.691i)T \)
7 \( 1 + (-0.450 + 0.892i)T \)
11 \( 1 + (0.778 + 0.628i)T \)
13 \( 1 + (0.660 + 0.750i)T \)
17 \( 1 + (-0.292 + 0.956i)T \)
19 \( 1 + (-0.967 + 0.251i)T \)
23 \( 1 + (0.660 - 0.750i)T \)
29 \( 1 + (-0.996 - 0.0848i)T \)
31 \( 1 + (0.660 - 0.750i)T \)
37 \( 1 + (0.778 + 0.628i)T \)
41 \( 1 + (0.210 - 0.977i)T \)
43 \( 1 + (0.985 - 0.169i)T \)
47 \( 1 + (-0.967 - 0.251i)T \)
53 \( 1 + (0.985 + 0.169i)T \)
59 \( 1 + (0.873 - 0.487i)T \)
61 \( 1 + (0.942 - 0.333i)T \)
67 \( 1 + (-0.127 - 0.991i)T \)
71 \( 1 + (-0.721 + 0.691i)T \)
73 \( 1 + (-0.594 + 0.803i)T \)
79 \( 1 + (-0.594 - 0.803i)T \)
83 \( 1 + (-0.594 + 0.803i)T \)
89 \( 1 + (-0.911 - 0.411i)T \)
97 \( 1 + (0.985 + 0.169i)T \)
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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

−28.08819565583633049138526475690, −26.884612150213340152052112036509, −25.50222726433818478758942621686, −24.76223087026146427854475123435, −23.86725646234070637577136582768, −23.13114845936693852802735049370, −22.54222234213515397487772476838, −20.98465611688572640427081694283, −19.911576307217916753945737706833, −19.35635634659655429719571517329, −17.626907011087199811475904340068, −16.68830506439789341230782998563, −15.9907265314710098808369961173, −14.57145863563590751042009176478, −13.28711490548792435968031533889, −13.00516412600161210965960037382, −11.67045149329024563200733186867, −11.01126204688430983392992753092, −8.743909222532210617443372895327, −7.638563574346799159390376973620, −6.76644380091697414161355886651, −5.63768730808650041656331841085, −4.27724343189851166736783938284, −3.09937517212758365821563868890, −1.06743682371427937295171605801, 2.34857508146198441495847715284, 3.72994808459838375328295163228, 4.36320329999006109316809224663, 5.97796676138258041285341160077, 6.69638797738689294666601304993, 8.71803936687021080387135817889, 9.991865112649748632297340380381, 11.08183384643845371838484167211, 11.777087849092685849062676117917, 12.82324916963932952506553959128, 14.5590897708174870305814918579, 15.02117310041325775572098229163, 15.85573283685142188294577361622, 16.94644175075548488421383939643, 18.7313948179372561663951548319, 19.489948868530031764668187024, 20.67860373716673995360940179970, 21.62367330704828960129282928788, 22.44336793194070138200492268765, 22.969347674908819649390941522776, 24.05657098251681015047404987885, 25.48097689896713743227806228081, 26.25457028329411322595013844529, 27.76110015310512484970867892257, 28.17081898515424599724680381568

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