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Properties

Label 1-193-193.39-r1-0-0
Degree $1$
Conductor $193$
Sign $0.999 - 0.0387i$
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  + (0.555 − 0.831i)2-s + (−0.923 + 0.382i)3-s + (−0.382 − 0.923i)4-s + (0.471 + 0.881i)5-s + (−0.195 + 0.980i)6-s + (0.707 − 0.707i)7-s + (−0.980 − 0.195i)8-s + (0.707 − 0.707i)9-s + (0.995 + 0.0980i)10-s + (−0.956 + 0.290i)11-s + (0.707 + 0.707i)12-s + (0.0980 + 0.995i)13-s + (−0.195 − 0.980i)14-s + (−0.773 − 0.634i)15-s + (−0.707 + 0.707i)16-s + (−0.471 + 0.881i)17-s + ⋯
L(s)  = 1  + (0.555 − 0.831i)2-s + (−0.923 + 0.382i)3-s + (−0.382 − 0.923i)4-s + (0.471 + 0.881i)5-s + (−0.195 + 0.980i)6-s + (0.707 − 0.707i)7-s + (−0.980 − 0.195i)8-s + (0.707 − 0.707i)9-s + (0.995 + 0.0980i)10-s + (−0.956 + 0.290i)11-s + (0.707 + 0.707i)12-s + (0.0980 + 0.995i)13-s + (−0.195 − 0.980i)14-s + (−0.773 − 0.634i)15-s + (−0.707 + 0.707i)16-s + (−0.471 + 0.881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $0.999 - 0.0387i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ 0.999 - 0.0387i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.828815549 - 0.03543701642i\)
\(L(\frac12)\) \(\approx\) \(1.828815549 - 0.03543701642i\)
\(L(1)\) \(\approx\) \(1.159126932 - 0.2252102774i\)
\(L(1)\) \(\approx\) \(1.159126932 - 0.2252102774i\)

Euler product

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

−26.91281890689182498807075655210, −25.311573303325363332663177664840, −24.75307406934299077849889520604, −24.10557305807375478524100886854, −23.16087582349912611032420651221, −22.246063382947012348030410722101, −21.29252814939285957823729951822, −20.58227843603862638434553964697, −18.60450713032281401943404375221, −17.803252313512244291939493480400, −17.23336487821332063802356554806, −15.97344795421279233020766746382, −15.51802787568786206565424927991, −13.83634363736857338848613319909, −13.1309160927559490257846379979, −12.22081617362084989804169802678, −11.32830780129686631150692744228, −9.6991043019744880709430703830, −8.29981795919087247429512262986, −7.573076481170536582367023272558, −5.969119292063898338465207431266, −5.38321180387751485067565096542, −4.661610310863309302645563498637, −2.57842222921062061958347290556, −0.71931822400937836727248220915, 1.1335619742698364595634457754, 2.56786054354579050397948381874, 4.09842866804015128826423836946, 4.9523972420144580709358777302, 6.13796968871598750291705520939, 7.18094297676344646687593483697, 9.21177886705776483652319172509, 10.546500688428514659030355739374, 10.69319933733335737243731476657, 11.754807818926619261275696319342, 12.96509024637304007053778378452, 14.01967500926315434686898250207, 14.87733042228504244301462646381, 16.01347295502834293447291245959, 17.5224026455363462975485470493, 18.05000461545919776983958616991, 19.075413463705777344353600058832, 20.4608278807970960684232702124, 21.33981194360248926572391470761, 21.862052229994642368044858854508, 22.98925573533066493294920079964, 23.532443089978992255480771950896, 24.456429939290245686840699598276, 26.462022729037432807117190195060, 26.690197841885195133150227196709

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