↑

Properties

Label 1-193-193.94-r1-0-0
Degree $1$
Conductor $193$
Sign $-0.857 - 0.514i$
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  + (0.555 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $-0.857 - 0.514i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ -0.857 - 0.514i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3377297028 + 1.220172000i\)
\(L(\frac12)\) \(\approx\) \(-0.3377297028 + 1.220172000i\)
\(L(1)\) \(\approx\) \(0.6408757085 + 0.7413818574i\)
\(L(1)\) \(\approx\) \(0.6408757085 + 0.7413818574i\)

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 \( 1 + (0.555 - 0.831i)T \)
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

−26.979910612261199149804322571600, −24.75862954032517139650331749174, −24.16828116108540269504401385463, −22.99131519672567078717873655415, −22.742057822367151603936849168156, −21.27727887840236583509629110142, −20.73141440472812860759313734729, −19.856579375220441485066630652143, −18.67535402924741359629873251435, −17.40524492530674656706499208600, −16.71829747646590035253526968904, −15.452971000836778622660884267759, −14.45734981737011305885743091263, −13.17262626247987640582042822008, −12.20657630830385668996887257608, −11.49428822971289091702792220943, −10.58506465897328232183546004264, −9.56032079558964061138450543940, −8.22034950218406787566712728550, −6.52426333311823171345525862134, −5.19821281383300387587145279541, −4.53145032632881879513479167558, −3.52811419329548524805061673809, −1.32124950937655938568176596771, −0.456268691812901311734655055962, 1.95682784499374530522666167395, 3.79632490653205405864379976925, 4.86881383948470716595835519296, 6.109816375559384456941046524315, 6.84690812808144361258071351156, 7.80220354092838065217059599982, 9.10422634426547148971155041606, 10.90531962053750757201244265895, 11.77728274735316058810669619146, 12.43364613752085344640144556781, 13.86902316796282299217050085843, 14.83241087282914727133078152525, 15.5215683626315749140747032296, 16.877730829888993991830959241664, 17.45677565494476739445068081119, 18.54359753142758514598464055696, 19.29894662792213299925263866544, 21.35197128501495721606573347164, 21.86575081010435628955408648046, 22.762238068371239229719050323043, 23.635109005204731952422332609423, 24.24744200874601787899907647283, 25.30097328797889482071487573605, 26.25665651489594782025302426287, 27.49743779767887669957053538158

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