↑

Properties

Label 1-193-193.89-r1-0-0
Degree $1$
Conductor $193$
Sign $0.981 + 0.189i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.980 + 0.195i)2-s + (−0.382 + 0.923i)3-s + (0.923 + 0.382i)4-s + (0.0980 − 0.995i)5-s + (−0.555 + 0.831i)6-s + (−0.707 − 0.707i)7-s + (0.831 + 0.555i)8-s + (−0.707 − 0.707i)9-s + (0.290 − 0.956i)10-s + (0.773 + 0.634i)11-s + (−0.707 + 0.707i)12-s + (0.956 − 0.290i)13-s + (−0.555 − 0.831i)14-s + (0.881 + 0.471i)15-s + (0.707 + 0.707i)16-s + (−0.0980 − 0.995i)17-s + ⋯
L(s)  = 1  + (0.980 + 0.195i)2-s + (−0.382 + 0.923i)3-s + (0.923 + 0.382i)4-s + (0.0980 − 0.995i)5-s + (−0.555 + 0.831i)6-s + (−0.707 − 0.707i)7-s + (0.831 + 0.555i)8-s + (−0.707 − 0.707i)9-s + (0.290 − 0.956i)10-s + (0.773 + 0.634i)11-s + (−0.707 + 0.707i)12-s + (0.956 − 0.290i)13-s + (−0.555 − 0.831i)14-s + (0.881 + 0.471i)15-s + (0.707 + 0.707i)16-s + (−0.0980 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $0.981 + 0.189i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ 0.981 + 0.189i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.218879395 + 0.3072965172i\)
\(L(\frac12)\) \(\approx\) \(3.218879395 + 0.3072965172i\)
\(L(1)\) \(\approx\) \(1.840023874 + 0.2691647042i\)
\(L(1)\) \(\approx\) \(1.840023874 + 0.2691647042i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−26.564359396744785558307619456398, −25.340935044888976244697637362083, −24.92587314546563797634553568181, −23.67363940619561792866597058535, −22.9649602130353006495497779858, −22.13178413483733720786364480152, −21.5359214342877858032646724958, −19.907059713497504323014876702760, −19.03968570629560416093441274859, −18.50879174231360497740802466037, −17.0552399936252177047507327439, −15.91510383757139523728914444116, −14.86037000197162786807805049588, −13.796486493498377217281387857191, −13.16544084893218601768405955936, −11.89410529745309047659870676754, −11.36413836556265113996475773080, −10.208406774509999693857426831983, −8.49512793516595718016796330630, −6.8652103426993257694117681354, −6.39014601070554412783016301766, −5.51710788201127223948845769056, −3.599851229186005296744073477551, −2.69099927311506980787215070452, −1.31802605830122143489793123402, 1.01735229399589752634328508263, 3.2099371891032016905584092906, 4.16095638928641645916552526222, 5.05259627097109443229610456671, 6.099322070835111950254588342428, 7.28641925068895349252532188267, 8.93913984034238030966921347704, 9.91291125293583206872219157081, 11.18063976499849374805306874821, 12.09286277885253497734254484263, 13.156135970948121342892492548639, 14.04145256045397070090535443187, 15.31180967214085177999420599043, 16.198649190077845351358709891783, 16.66740228456472378321198743776, 17.706501612059468028636201722706, 19.82839814971020914416075908778, 20.41574614007772362206287071096, 21.07124302161231908182527937559, 22.337851931421214596084501946749, 22.889145847660279178345315116873, 23.696026989305563835639469195038, 24.9778505969482664395638669302, 25.63818152447427182569778391099, 26.81805125055234578115444872933

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