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Properties

Label 1-197-197.44-r1-0-0
Degree $1$
Conductor $197$
Sign $-0.647 + 0.761i$
Analytic cond. $21.1705$
Root an. cond. $21.1705$
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  + (0.545 + 0.838i)2-s + (−0.695 − 0.718i)3-s + (−0.404 + 0.914i)4-s + (0.886 + 0.462i)5-s + (0.222 − 0.974i)6-s + (0.838 + 0.545i)7-s + (−0.987 + 0.159i)8-s + (−0.0320 + 0.999i)9-s + (0.0960 + 0.995i)10-s + (−0.855 − 0.518i)11-s + (0.938 − 0.345i)12-s + (0.958 − 0.284i)13-s + i·14-s + (−0.284 − 0.958i)15-s + (−0.672 − 0.740i)16-s + (0.598 + 0.801i)17-s + ⋯
L(s)  = 1  + (0.545 + 0.838i)2-s + (−0.695 − 0.718i)3-s + (−0.404 + 0.914i)4-s + (0.886 + 0.462i)5-s + (0.222 − 0.974i)6-s + (0.838 + 0.545i)7-s + (−0.987 + 0.159i)8-s + (−0.0320 + 0.999i)9-s + (0.0960 + 0.995i)10-s + (−0.855 − 0.518i)11-s + (0.938 − 0.345i)12-s + (0.958 − 0.284i)13-s + i·14-s + (−0.284 − 0.958i)15-s + (−0.672 − 0.740i)16-s + (0.598 + 0.801i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.647 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.647 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(197\)
Sign: $-0.647 + 0.761i$
Analytic conductor: \(21.1705\)
Root analytic conductor: \(21.1705\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{197} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 197,\ (1:\ ),\ -0.647 + 0.761i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8522959617 + 1.842848933i\)
\(L(\frac12)\) \(\approx\) \(0.8522959617 + 1.842848933i\)
\(L(1)\) \(\approx\) \(1.092085713 + 0.7148029936i\)
\(L(1)\) \(\approx\) \(1.092085713 + 0.7148029936i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad197 \( 1 \)
good2 \( 1 + (0.545 + 0.838i)T \)
3 \( 1 + (-0.695 - 0.718i)T \)
5 \( 1 + (0.886 + 0.462i)T \)
7 \( 1 + (0.838 + 0.545i)T \)
11 \( 1 + (-0.855 - 0.518i)T \)
13 \( 1 + (0.958 - 0.284i)T \)
17 \( 1 + (0.598 + 0.801i)T \)
19 \( 1 + (-0.623 + 0.781i)T \)
23 \( 1 + (0.991 - 0.127i)T \)
29 \( 1 + (-0.345 - 0.938i)T \)
31 \( 1 + (-0.315 + 0.949i)T \)
37 \( 1 + (-0.672 + 0.740i)T \)
41 \( 1 + (-0.801 + 0.598i)T \)
43 \( 1 + (-0.518 + 0.855i)T \)
47 \( 1 + (-0.926 + 0.375i)T \)
53 \( 1 + (-0.761 + 0.648i)T \)
59 \( 1 + (-0.0960 + 0.995i)T \)
61 \( 1 + (0.718 + 0.695i)T \)
67 \( 1 + (-0.375 - 0.926i)T \)
71 \( 1 + (0.999 + 0.0320i)T \)
73 \( 1 + (0.740 + 0.672i)T \)
79 \( 1 + (0.886 - 0.462i)T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (-0.315 - 0.949i)T \)
97 \( 1 + (0.981 - 0.191i)T \)
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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

−26.63802314400280539495249274097, −25.47743668042770565047979427465, −23.955288767896818277295635463974, −23.48615731899361339231792059524, −22.459545184690609423726944248908, −21.35870020907437874172593524797, −20.86608138138665732942745597680, −20.34313425324157559690693988909, −18.53695664511905207093384126532, −17.79695181348891744976392593707, −16.85076108982681162624318009742, −15.605741354499748280247326509985, −14.537264665514480711053785397459, −13.50518513920021197606715855363, −12.62602591643476709778893293007, −11.32817380215234585092046489331, −10.69667957264026225185992443187, −9.752380563777443945153302903922, −8.7666519870188693617687556924, −6.728176839830113457764415612993, −5.24113991243822010762137663709, −4.980266804112118492186902827124, −3.620328161790926132393986875168, −1.952121467436539505333693471781, −0.6528912105179742010612249745, 1.60183096489217865887855166404, 3.06816446289142830573457026366, 4.98100652534855924683101306120, 5.78993503348002898466725195261, 6.43782477462389368980928635119, 7.83600494049011291503589010056, 8.55962992974394254739603107069, 10.43800043314904579276274862282, 11.40949998651755132584560311486, 12.68875650882599174610360280089, 13.37534670561919365834203723680, 14.34848289404098987478928110717, 15.33175529953188460548905686652, 16.62056388228581740739883686967, 17.38912514934850538908642422758, 18.31161724615146809616778265956, 18.759814678876493217762671990527, 21.14181684187409126592447041033, 21.33257024115895899971451108667, 22.607174867809703446445028471424, 23.3652183972799879641127655605, 24.19475494002121210801190757729, 25.12827171470066723446655772960, 25.66933930771671585431477640009, 26.9227165422100947868658818631

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