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Properties

Label 1-193-193.119-r1-0-0
Degree $1$
Conductor $193$
Sign $0.247 - 0.968i$
Analytic cond. $20.7407$
Root an. cond. $20.7407$
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  + (0.831 − 0.555i)2-s + (0.923 + 0.382i)3-s + (0.382 − 0.923i)4-s + (−0.290 − 0.956i)5-s + (0.980 − 0.195i)6-s + (0.707 + 0.707i)7-s + (−0.195 − 0.980i)8-s + (0.707 + 0.707i)9-s + (−0.773 − 0.634i)10-s + (−0.471 − 0.881i)11-s + (0.707 − 0.707i)12-s + (0.634 + 0.773i)13-s + (0.980 + 0.195i)14-s + (0.0980 − 0.995i)15-s + (−0.707 − 0.707i)16-s + (0.290 − 0.956i)17-s + ⋯
L(s)  = 1  + (0.831 − 0.555i)2-s + (0.923 + 0.382i)3-s + (0.382 − 0.923i)4-s + (−0.290 − 0.956i)5-s + (0.980 − 0.195i)6-s + (0.707 + 0.707i)7-s + (−0.195 − 0.980i)8-s + (0.707 + 0.707i)9-s + (−0.773 − 0.634i)10-s + (−0.471 − 0.881i)11-s + (0.707 − 0.707i)12-s + (0.634 + 0.773i)13-s + (0.980 + 0.195i)14-s + (0.0980 − 0.995i)15-s + (−0.707 − 0.707i)16-s + (0.290 − 0.956i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $0.247 - 0.968i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ 0.247 - 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.441271144 - 2.673694415i\)
\(L(\frac12)\) \(\approx\) \(3.441271144 - 2.673694415i\)
\(L(1)\) \(\approx\) \(2.182290174 - 0.9683204799i\)
\(L(1)\) \(\approx\) \(2.182290174 - 0.9683204799i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad193 \( 1 \)
good2 \( 1 + (0.831 - 0.555i)T \)
3 \( 1 + (0.923 + 0.382i)T \)
5 \( 1 + (-0.290 - 0.956i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (-0.471 - 0.881i)T \)
13 \( 1 + (0.634 + 0.773i)T \)
17 \( 1 + (0.290 - 0.956i)T \)
19 \( 1 + (0.956 - 0.290i)T \)
23 \( 1 + (0.831 - 0.555i)T \)
29 \( 1 + (-0.995 + 0.0980i)T \)
31 \( 1 + (-0.555 - 0.831i)T \)
37 \( 1 + (-0.995 - 0.0980i)T \)
41 \( 1 + (0.471 + 0.881i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.773 + 0.634i)T \)
53 \( 1 + (0.634 + 0.773i)T \)
59 \( 1 + (0.923 + 0.382i)T \)
61 \( 1 + (0.956 + 0.290i)T \)
67 \( 1 + (-0.555 - 0.831i)T \)
71 \( 1 + (-0.956 + 0.290i)T \)
73 \( 1 + (0.995 - 0.0980i)T \)
79 \( 1 + (-0.881 + 0.471i)T \)
83 \( 1 + (0.195 - 0.980i)T \)
89 \( 1 + (-0.634 + 0.773i)T \)
97 \( 1 + (0.831 + 0.555i)T \)
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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

−26.54864121401535575610535797828, −25.95350195373229176793011612082, −25.16782346229511063955735123961, −24.054539439696234719423247272984, −23.32131006283810278787616180695, −22.54197779874359973418506986528, −21.13501700331673200845366484928, −20.55725230731410282077220614588, −19.51915492314032934843746988219, −18.1504794049131270229008645207, −17.52515744490241224142449756129, −15.87819951393799351462363684385, −14.99058685475791446533260305133, −14.45762654769862316355229544178, −13.47320675292288325418792344320, −12.61122666695835323571235980507, −11.28471325013693574078110116656, −10.18373028493340853738358711000, −8.40300010228117429589032523247, −7.51482116963221718570333181740, −7.02213594661039657515958374543, −5.465328769566195633833414605179, −3.92830129529839575935372089900, −3.21405253274547926880627813173, −1.79046392500730586302504233994, 1.15939132772181598814486292689, 2.4735708745118947007073955171, 3.65497144121505341926770411998, 4.7962259785938127061797302014, 5.567109544144253616210676525753, 7.49097074177997417372594389625, 8.77781230755583626622940294710, 9.4035957506401596673469613503, 10.9986333016162588906145120721, 11.76940675999529539719295114613, 13.03345419434462675773189722847, 13.78000482636166100455433349178, 14.7244986673784462046064775894, 15.79332316674958424032751802367, 16.366567703387797632860847006367, 18.470028463720807965146809819342, 19.09388920683625070928975989593, 20.34919360788130313628071026004, 20.88765680369954661585206128919, 21.46605315247617895900139334595, 22.63210525341128704302082012838, 24.192376324373809227138227722329, 24.29368752179507212582083260895, 25.393550825697981656677200471368, 26.78216371064808828041326732967

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