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Properties

Label 1-199-199.130-r0-0-0
Degree $1$
Conductor $199$
Sign $0.993 + 0.115i$
Analytic cond. $0.924152$
Root an. cond. $0.924152$
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  + (−0.266 − 0.963i)2-s + (0.805 + 0.592i)3-s + (−0.857 + 0.513i)4-s + (0.928 − 0.371i)5-s + (0.356 − 0.934i)6-s + (−0.553 + 0.832i)7-s + (0.723 + 0.690i)8-s + (0.296 + 0.954i)9-s + (−0.605 − 0.795i)10-s + (0.841 + 0.540i)11-s + (−0.995 − 0.0950i)12-s + (−0.701 + 0.712i)13-s + (0.950 + 0.312i)14-s + (0.967 + 0.251i)15-s + (0.472 − 0.881i)16-s + (−0.888 + 0.458i)17-s + ⋯
L(s)  = 1  + (−0.266 − 0.963i)2-s + (0.805 + 0.592i)3-s + (−0.857 + 0.513i)4-s + (0.928 − 0.371i)5-s + (0.356 − 0.934i)6-s + (−0.553 + 0.832i)7-s + (0.723 + 0.690i)8-s + (0.296 + 0.954i)9-s + (−0.605 − 0.795i)10-s + (0.841 + 0.540i)11-s + (−0.995 − 0.0950i)12-s + (−0.701 + 0.712i)13-s + (0.950 + 0.312i)14-s + (0.967 + 0.251i)15-s + (0.472 − 0.881i)16-s + (−0.888 + 0.458i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.993 + 0.115i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.993 + 0.115i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.316082790 + 0.07651384190i\)
\(L(\frac12)\) \(\approx\) \(1.316082790 + 0.07651384190i\)
\(L(1)\) \(\approx\) \(1.187021566 - 0.09455141368i\)
\(L(1)\) \(\approx\) \(1.187021566 - 0.09455141368i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (-0.266 - 0.963i)T \)
3 \( 1 + (0.805 + 0.592i)T \)
5 \( 1 + (0.928 - 0.371i)T \)
7 \( 1 + (-0.553 + 0.832i)T \)
11 \( 1 + (0.841 + 0.540i)T \)
13 \( 1 + (-0.701 + 0.712i)T \)
17 \( 1 + (-0.888 + 0.458i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (0.873 - 0.486i)T \)
29 \( 1 + (0.902 + 0.429i)T \)
31 \( 1 + (-0.916 - 0.400i)T \)
37 \( 1 + (0.766 + 0.642i)T \)
41 \( 1 + (0.678 - 0.734i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.444 - 0.895i)T \)
53 \( 1 + (0.967 - 0.251i)T \)
59 \( 1 + (-0.327 - 0.945i)T \)
61 \( 1 + (-0.142 - 0.989i)T \)
67 \( 1 + (-0.786 - 0.618i)T \)
71 \( 1 + (0.950 - 0.312i)T \)
73 \( 1 + (-0.0158 + 0.999i)T \)
79 \( 1 + (0.527 + 0.849i)T \)
83 \( 1 + (-0.995 + 0.0950i)T \)
89 \( 1 + (-0.0792 - 0.996i)T \)
97 \( 1 + (0.110 + 0.993i)T \)
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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

−26.713988782511137943509164670909, −25.76467647720763760103663765270, −25.076542017866241347084731600163, −24.487573109009840568841765763368, −23.2814758533881487972068744922, −22.430355593035217967311102033743, −21.28285566948351785689396294382, −19.75853717871371002282474120629, −19.329893359438653001600083805887, −18.03593096698277365630044881976, −17.40469073906713675368217641084, −16.4221824984353946755841282055, −15.005835088465249889569689083683, −14.3185628648923490292632614380, −13.46588720357087248955707296673, −12.81074879078356916630228745654, −10.72219995422927983591124546383, −9.60642640625334753148038363272, −8.93281551733217982991361124117, −7.58270232231384979608744691241, −6.7577005790850897327795115885, −5.978944813727136750578450249563, −4.25031902075263660609289663528, −2.8205723293963377698290271956, −1.1260544827816833001522727713, 1.937015784091466633383412684545, 2.561970546934297433208954373755, 4.08495241902092446944981681373, 5.02030488388049782185631665642, 6.73004148340581090372862711143, 8.61471365786566950344365193472, 9.14404277148795543896574493546, 9.808394022699865430786545771640, 10.92525619056455797281720384047, 12.38172473475827880776529193901, 13.07483864510912621587052004062, 14.19145919711525434234050661464, 15.079358473401706926788623895825, 16.58999793347417018371338501928, 17.343559037259667237122549467148, 18.618353839940201842169430664010, 19.53808664767211568364122989278, 20.18793620399254406359000700528, 21.40997159659263250439200043662, 21.74756956671555704932356344329, 22.57436103441673447039079213907, 24.3900554750327843163182947387, 25.444040864422524202305681659449, 25.94208197147447874604367478999, 27.066965004104924784474641560928

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