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Properties

Label 1-193-193.71-r1-0-0
Degree $1$
Conductor $193$
Sign $0.131 - 0.991i$
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 + (0.881 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $0.131 - 0.991i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ 0.131 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6048870897 - 0.5300527908i\)
\(L(\frac12)\) \(\approx\) \(0.6048870897 - 0.5300527908i\)
\(L(1)\) \(\approx\) \(0.6885176212 + 0.02286763709i\)
\(L(1)\) \(\approx\) \(0.6885176212 + 0.02286763709i\)

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 \( 1 + (0.881 - 0.471i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (-0.290 - 0.956i)T \)
13 \( 1 + (0.995 - 0.0980i)T \)
17 \( 1 + (-0.881 - 0.471i)T \)
19 \( 1 + (-0.471 - 0.881i)T \)
23 \( 1 + (-0.555 + 0.831i)T \)
29 \( 1 + (-0.773 + 0.634i)T \)
31 \( 1 + (-0.831 - 0.555i)T \)
37 \( 1 + (-0.773 - 0.634i)T \)
41 \( 1 + (0.290 + 0.956i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (-0.0980 - 0.995i)T \)
53 \( 1 + (0.995 - 0.0980i)T \)
59 \( 1 + (-0.923 + 0.382i)T \)
61 \( 1 + (-0.471 + 0.881i)T \)
67 \( 1 + (-0.831 - 0.555i)T \)
71 \( 1 + (0.471 + 0.881i)T \)
73 \( 1 + (0.773 - 0.634i)T \)
79 \( 1 + (0.956 - 0.290i)T \)
83 \( 1 + (-0.980 + 0.195i)T \)
89 \( 1 + (-0.995 - 0.0980i)T \)
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

−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