Properties

Degree 1
Conductor 193
Sign $-0.857 + 0.514i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(193\)
\( \varepsilon \)  =  $-0.857 + 0.514i$
motivic weight  =  \(0\)
character  :  $\chi_{193} (154, \cdot )$
Sato-Tate  :  $\mu(64)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 193,\ (1:\ ),\ -0.857 + 0.514i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.3377297028 - 1.220172000i$
$L(\frac12,\chi)$  $\approx$  $-0.3377297028 - 1.220172000i$
$L(\chi,1)$  $\approx$  0.6408757085 - 0.7413818574i
$L(1,\chi)$  $\approx$  0.6408757085 - 0.7413818574i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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