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Properties

Label 1-199-199.26-r0-0-0
Degree $1$
Conductor $199$
Sign $-0.588 + 0.808i$
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  + (0.472 − 0.881i)2-s + (−0.823 + 0.567i)3-s + (−0.553 − 0.832i)4-s + (0.0475 − 0.998i)5-s + (0.110 + 0.993i)6-s + (−0.701 + 0.712i)7-s + (−0.995 + 0.0950i)8-s + (0.356 − 0.934i)9-s + (−0.857 − 0.513i)10-s + (−0.654 + 0.755i)11-s + (0.928 + 0.371i)12-s + (−0.999 + 0.0317i)13-s + (0.296 + 0.954i)14-s + (0.527 + 0.849i)15-s + (−0.386 + 0.922i)16-s + (−0.327 − 0.945i)17-s + ⋯
L(s)  = 1  + (0.472 − 0.881i)2-s + (−0.823 + 0.567i)3-s + (−0.553 − 0.832i)4-s + (0.0475 − 0.998i)5-s + (0.110 + 0.993i)6-s + (−0.701 + 0.712i)7-s + (−0.995 + 0.0950i)8-s + (0.356 − 0.934i)9-s + (−0.857 − 0.513i)10-s + (−0.654 + 0.755i)11-s + (0.928 + 0.371i)12-s + (−0.999 + 0.0317i)13-s + (0.296 + 0.954i)14-s + (0.527 + 0.849i)15-s + (−0.386 + 0.922i)16-s + (−0.327 − 0.945i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $-0.588 + 0.808i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ -0.588 + 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03403130589 - 0.06681840322i\)
\(L(\frac12)\) \(\approx\) \(-0.03403130589 - 0.06681840322i\)
\(L(1)\) \(\approx\) \(0.5449986802 - 0.2654772429i\)
\(L(1)\) \(\approx\) \(0.5449986802 - 0.2654772429i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.472 - 0.881i)T \)
3 \( 1 + (-0.823 + 0.567i)T \)
5 \( 1 + (0.0475 - 0.998i)T \)
7 \( 1 + (-0.701 + 0.712i)T \)
11 \( 1 + (-0.654 + 0.755i)T \)
13 \( 1 + (-0.999 + 0.0317i)T \)
17 \( 1 + (-0.327 - 0.945i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (-0.444 - 0.895i)T \)
29 \( 1 + (-0.204 + 0.978i)T \)
31 \( 1 + (-0.0792 + 0.996i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (-0.987 + 0.158i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (-0.266 - 0.963i)T \)
53 \( 1 + (0.527 - 0.849i)T \)
59 \( 1 + (0.235 - 0.971i)T \)
61 \( 1 + (0.841 - 0.540i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (0.296 - 0.954i)T \)
73 \( 1 + (0.997 + 0.0634i)T \)
79 \( 1 + (-0.605 - 0.795i)T \)
83 \( 1 + (0.928 - 0.371i)T \)
89 \( 1 + (0.950 - 0.312i)T \)
97 \( 1 + (0.902 - 0.429i)T \)
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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

−27.188104039869079515019305076415, −26.37173470162585663891936720230, −25.73862678433147992383248312130, −24.350307571361432188319476746398, −23.803237000704087652281953325499, −22.88568225806304714887009980246, −22.16539055712697254018989661020, −21.51959682907865559916214885098, −19.5729549414837687057688093665, −18.7578920437680596781027173669, −17.62333558481034974146157315375, −17.06028127966772608519792259662, −15.95372688420515594138550141134, −15.05252907849202784974923903698, −13.624375428925367613027711670319, −13.29379372109384500251870008367, −11.968971049978095167371684893703, −10.87584992659499503622905801970, −9.78501943858934299915638125283, −7.932408472969759293015130118451, −7.15266443080799948708707149001, −6.34717761710599839981247560691, −5.42719540360956119922318221594, −3.968110900179859104521166876902, −2.63103079646095196374554260365, 0.05092049587488900405022201817, 1.976585042203733964849844079432, 3.49102444945472385669717447973, 5.01346967073774581766086187594, 5.15819969702832067782256854476, 6.66180626356991426407209609256, 8.72628171084719820799258105069, 9.7506256528126599062798090495, 10.29145609659914151928564517177, 11.96684101866968779980737279167, 12.22362315289366464824378255267, 13.148568194247380526856889954333, 14.66687960188438184393469910014, 15.71013937777053982951469665786, 16.52488742433460850568011358079, 17.810835076044201922653512587740, 18.63226370423707391274674700115, 20.02797245034997127902635872654, 20.64384795576798070275856325594, 21.64212689535742429402746412244, 22.39132245171117839151025478664, 23.17345065075412663410779834697, 24.15861549031963654913585051304, 25.1491690435380930540293982402, 26.728598432445759382264045544829

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