Properties

Degree 1
Conductor 193
Sign $0.131 - 0.991i$
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 + (0.881 − 0.471i)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 + (−0.0980 + 0.995i)10-s + (−0.290 − 0.956i)11-s + (0.707 + 0.707i)12-s + (0.995 − 0.0980i)13-s + (0.195 + 0.980i)14-s + (−0.634 + 0.773i)15-s + (−0.707 + 0.707i)16-s + (−0.881 − 0.471i)17-s + ⋯
L(s,χ)  = 1  + (−0.555 + 0.831i)2-s + (−0.923 + 0.382i)3-s + (−0.382 − 0.923i)4-s + (0.881 − 0.471i)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 + (−0.0980 + 0.995i)10-s + (−0.290 − 0.956i)11-s + (0.707 + 0.707i)12-s + (0.995 − 0.0980i)13-s + (0.195 + 0.980i)14-s + (−0.634 + 0.773i)15-s + (−0.707 + 0.707i)16-s + (−0.881 − 0.471i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.131 - 0.991i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.131 - 0.991i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(193\)
\( \varepsilon \)  =  $0.131 - 0.991i$
motivic weight  =  \(0\)
character  :  $\chi_{193} (71, \cdot )$
Sato-Tate  :  $\mu(64)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 193,\ (1:\ ),\ 0.131 - 0.991i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6048870897 - 0.5300527908i$
$L(\frac12,\chi)$  $\approx$  $0.6048870897 - 0.5300527908i$
$L(\chi,1)$  $\approx$  0.6885176212 + 0.02286763709i
$L(1,\chi)$  $\approx$  0.6885176212 + 0.02286763709i

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

−27.39896409998224384054553976662, −26.078986594399417325390064744408, −25.33061421869796380060692132727, −24.22161136678675655591390191917, −22.85209927002359945863544962268, −22.20810745043245666835687373831, −21.25615979571720965567574581724, −20.5614294123712714051234957982, −18.921176266438553314432368320497, −18.283713412364197843125539360291, −17.70952993216828579683331490756, −16.89085328579778842647702547039, −15.508893425301400476163741429141, −14.02464412270969816924865281601, −12.88973362646130173059353901202, −12.16254131400452867544243009555, −10.93527802482751062044989403454, −10.47256780120208472025427957459, −9.16482272699777935262091596095, −7.97016231736063600623407151916, −6.6627515470384117391360447958, −5.523627459954324742572853937758, −4.20198300621560525799015443508, −2.19888675766252583492688367301, −1.63940688776578058483040757842, 0.389990859631042869743797969636, 1.53217933429391388157010989582, 4.19412578985308556259676900379, 5.290926432267633952556487967059, 6.020858263814071570347269217631, 7.15271540256848817245526753326, 8.5780440977654640397593845090, 9.45522956015441675735995703937, 10.73725484572616469422232904882, 11.16655479004720052722735503469, 13.19126998934672534321597505627, 13.81316932963107029941246769521, 15.19856147811883894461192013712, 16.26751967658762937297932093989, 16.82174540867977764931414938753, 17.90706548972313885782823766285, 18.16792272185471092070367401346, 19.88192125121077047820206487714, 20.99823120587365259666038559690, 21.870865753418335211369743541865, 23.05365746801218181161724944758, 24.0375261882518553268980130929, 24.38638584271553926481252739045, 25.85356243082675233540069202817, 26.529202107745171708383298133347

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