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Properties

Label 1-193-193.33-r1-0-0
Degree $1$
Conductor $193$
Sign $0.170 + 0.985i$
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  + (−0.195 + 0.980i)2-s + (0.382 − 0.923i)3-s + (−0.923 − 0.382i)4-s + (−0.773 + 0.634i)5-s + (0.831 + 0.555i)6-s + (−0.707 − 0.707i)7-s + (0.555 − 0.831i)8-s + (−0.707 − 0.707i)9-s + (−0.471 − 0.881i)10-s + (−0.995 + 0.0980i)11-s + (−0.707 + 0.707i)12-s + (0.881 + 0.471i)13-s + (0.831 − 0.555i)14-s + (0.290 + 0.956i)15-s + (0.707 + 0.707i)16-s + (0.773 + 0.634i)17-s + ⋯
L(s)  = 1  + (−0.195 + 0.980i)2-s + (0.382 − 0.923i)3-s + (−0.923 − 0.382i)4-s + (−0.773 + 0.634i)5-s + (0.831 + 0.555i)6-s + (−0.707 − 0.707i)7-s + (0.555 − 0.831i)8-s + (−0.707 − 0.707i)9-s + (−0.471 − 0.881i)10-s + (−0.995 + 0.0980i)11-s + (−0.707 + 0.707i)12-s + (0.881 + 0.471i)13-s + (0.831 − 0.555i)14-s + (0.290 + 0.956i)15-s + (0.707 + 0.707i)16-s + (0.773 + 0.634i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $0.170 + 0.985i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ 0.170 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7319910550 + 0.6162898753i\)
\(L(\frac12)\) \(\approx\) \(0.7319910550 + 0.6162898753i\)
\(L(1)\) \(\approx\) \(0.7475145199 + 0.2008393836i\)
\(L(1)\) \(\approx\) \(0.7475145199 + 0.2008393836i\)

Euler product

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

−26.83847696251485549713088165056, −25.921333184555568935729162161868, −24.99277070018629762446388723928, −23.17288143408033182566622662677, −22.81786300545905872073845440432, −21.42906389178153027982953445641, −20.86772252497887833071454946054, −20.06583901975743308044319740902, −19.08846142928583381250093658098, −18.32771324814950756455162490613, −16.66301322058062752484538131679, −16.01665171258848222566722740693, −15.022027172485443658541165743359, −13.60346099954005633951891797570, −12.64044955052201500052855132122, −11.713973090290214919557774854959, −10.529206281522481646224697012380, −9.751329600627773421753049770773, −8.51110963903396230261819116777, −8.13035750415839176053350912095, −5.65012805654899150794805588246, −4.56428302305251232933445173173, −3.46915701011499327861345427299, −2.57580174505736174367853310587, −0.46581916266951831902131707539, 0.91913649466415748769893625396, 3.02196008303750407019090458228, 4.185351105230404133526059541786, 6.03109030148990085304689375264, 6.841290680436544208584188158840, 7.705334825070483334893380337161, 8.45845323702783985785664245397, 9.891471869989254724350255976001, 11.07134456953146074040843115741, 12.63972261588590084689948224979, 13.46071980978536359447295761611, 14.32432652061318506797240081752, 15.41284796163620908483089557969, 16.20941631223661023928660092222, 17.47829074579433658972527600111, 18.36864506829100812296822708337, 19.21892910279596726919334830607, 19.76214731295961487677976117135, 21.400599294311519287527636409171, 23.01661951577465244224919992478, 23.37627573300854277143612521865, 23.924535462507547405412049666652, 25.37123438527251943968226844041, 26.0884138741243548313281860381, 26.43885431361330873247632200726

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