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Properties

Label 1-199-199.116-r0-0-0
Degree $1$
Conductor $199$
Sign $0.841 + 0.539i$
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  + (0.981 − 0.189i)2-s + (−0.888 + 0.458i)3-s + (0.928 − 0.371i)4-s + (−0.959 − 0.281i)5-s + (−0.786 + 0.618i)6-s + (0.0475 + 0.998i)7-s + (0.841 − 0.540i)8-s + (0.580 − 0.814i)9-s + (−0.995 − 0.0950i)10-s + (0.415 + 0.909i)11-s + (−0.654 + 0.755i)12-s + (0.981 − 0.189i)13-s + (0.235 + 0.971i)14-s + (0.981 − 0.189i)15-s + (0.723 − 0.690i)16-s + (0.415 + 0.909i)17-s + ⋯
L(s)  = 1  + (0.981 − 0.189i)2-s + (−0.888 + 0.458i)3-s + (0.928 − 0.371i)4-s + (−0.959 − 0.281i)5-s + (−0.786 + 0.618i)6-s + (0.0475 + 0.998i)7-s + (0.841 − 0.540i)8-s + (0.580 − 0.814i)9-s + (−0.995 − 0.0950i)10-s + (0.415 + 0.909i)11-s + (−0.654 + 0.755i)12-s + (0.981 − 0.189i)13-s + (0.235 + 0.971i)14-s + (0.981 − 0.189i)15-s + (0.723 − 0.690i)16-s + (0.415 + 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.841 + 0.539i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.841 + 0.539i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.422043906 + 0.4169368208i\)
\(L(\frac12)\) \(\approx\) \(1.422043906 + 0.4169368208i\)
\(L(1)\) \(\approx\) \(1.337454144 + 0.1685597508i\)
\(L(1)\) \(\approx\) \(1.337454144 + 0.1685597508i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.981 - 0.189i)T \)
3 \( 1 + (-0.888 + 0.458i)T \)
5 \( 1 + (-0.959 - 0.281i)T \)
7 \( 1 + (0.0475 + 0.998i)T \)
11 \( 1 + (0.415 + 0.909i)T \)
13 \( 1 + (0.981 - 0.189i)T \)
17 \( 1 + (0.415 + 0.909i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.928 + 0.371i)T \)
29 \( 1 + (-0.327 - 0.945i)T \)
31 \( 1 + (-0.888 - 0.458i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.580 - 0.814i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.0475 + 0.998i)T \)
53 \( 1 + (0.981 + 0.189i)T \)
59 \( 1 + (-0.142 - 0.989i)T \)
61 \( 1 + (-0.959 + 0.281i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (0.235 - 0.971i)T \)
73 \( 1 + (0.928 + 0.371i)T \)
79 \( 1 + (0.723 - 0.690i)T \)
83 \( 1 + (-0.654 - 0.755i)T \)
89 \( 1 + (-0.327 - 0.945i)T \)
97 \( 1 + (-0.888 - 0.458i)T \)
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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

−26.89606328961977936203379038599, −25.72772513043691819222624737877, −24.47895733551566028850835490585, −23.70262421674870763573540885752, −23.21409779281887540706507413857, −22.44581356016419311943513510327, −21.411416213530462812777898382899, −20.22053147947699744127718440721, −19.29652969821499928516913564920, −18.22128341150149943230557202635, −16.62257049939807290767476139752, −16.47879444799520363706096165549, −15.21364088454752374348000971482, −13.9484362641472376105294960682, −13.22473748009729501064273482722, −12.06935838382892414042189655048, −11.10421605387827680888441256254, −10.79982567202617077915677669652, −8.417728659382009630600962701894, −7.113017356773096774285389760, −6.7316276456978473388277453701, −5.30293461894819557328427421259, −4.207905447649263789849407605330, −3.18328644457423958426621413102, −1.09598752174782563229008141053, 1.60206447582330396618311314625, 3.53623216820614222396679731819, 4.315319405911276611953656543589, 5.47792950279902174205734388785, 6.30120382353587050698081285927, 7.67615271765835057406847433935, 9.21278586134881703646327906624, 10.625723517553442883776179576946, 11.44839162872463902720065731826, 12.3289284354821447284443303613, 12.83114887891377602310661458928, 14.78757666401765220839665632807, 15.26754111676481023377706334160, 16.114183054639594108775673403935, 17.09397336003287460560417001110, 18.54559743087668990156321623998, 19.49534125588099175228019207745, 20.80035239292519675043443785982, 21.29084215629884991690979392749, 22.59195857979179627438369050469, 22.98214765848799580969822633061, 23.8443570780280217708919329553, 24.84670623431741348787018744505, 25.87401510228994345315668270959, 27.550540920543168898303758800127

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