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Properties

Label 1-149-149.22-r0-0-0
Degree $1$
Conductor $149$
Sign $0.819 + 0.572i$
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  + (−0.0424 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.819 + 0.572i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.819 + 0.572i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3645947616 + 0.1146725363i\)
\(L(\frac12)\) \(\approx\) \(0.3645947616 + 0.1146725363i\)
\(L(1)\) \(\approx\) \(0.5404099637 - 0.1925935399i\)
\(L(1)\) \(\approx\) \(0.5404099637 - 0.1925935399i\)

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 \( 1 + (0.721 - 0.691i)T \)
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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

−27.68289218390705247646380228668, −27.26925604929209213745923847828, −25.99886511601274363789712244895, −25.10458499851191399414929967499, −24.16974208916836030149687297278, −23.00954958174054932917892109044, −22.41332381549277242789473351511, −21.42110490714242744908494083585, −20.026018275650082079708620405422, −19.051801814157607795157291341861, −17.333305608513708399425314164486, −16.987373895808866603825390297550, −15.85858936750596169011603464462, −15.39935008173944481486502345144, −14.03257632618122811133721079230, −12.6742737208536145191660409017, −11.82764269172043972537463439423, −9.899432818398270990815857099179, −9.38855003213944148616836231276, −8.24017361323872911401713798035, −6.68227307960750399312001432291, −5.63870097644441631746472585448, −4.64049383192847034666485750302, −3.56214548239017880535862929789, −0.362489210895175332695479109451, 1.687768316587517470494463686419, 3.0229206989895181407309789306, 4.26836134920319220349611672637, 6.14202368232526897702388207, 7.0043454688652167044317023730, 8.4260385854741415575044893655, 9.97385968351187974950592296497, 10.74736261619335540854625362955, 12.03046867878943119687683184054, 12.46619964183055839348135467794, 13.85954248037759942916692200756, 14.61216203722718143149201948028, 16.57605979097224052250024225000, 17.42380928507445858229772017638, 18.57411037753967153976337591374, 19.3915569914240548604674269802, 19.69619693373268609343730493588, 21.58762910114244997023971325476, 22.37273322468661358634987194875, 23.016344648573777866816154996722, 23.92792784020303354480743892414, 25.47259955383291574944720976732, 26.374377285184333559513684136564, 27.47601551758817352188361970004, 28.41435090243968072569922554982

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