Properties

Label 1-193-193.164-r1-0-0
Degree $1$
Conductor $193$
Sign $-0.336 + 0.941i$
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 + (0.634 + 0.773i)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 + (0.881 − 0.471i)10-s + (−0.0980 − 0.995i)11-s + (−0.707 + 0.707i)12-s + (0.471 − 0.881i)13-s + (−0.831 + 0.555i)14-s + (0.956 − 0.290i)15-s + (0.707 + 0.707i)16-s + (−0.634 + 0.773i)17-s + ⋯
L(s)  = 1  + (0.195 − 0.980i)2-s + (0.382 − 0.923i)3-s + (−0.923 − 0.382i)4-s + (0.634 + 0.773i)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 + (0.881 − 0.471i)10-s + (−0.0980 − 0.995i)11-s + (−0.707 + 0.707i)12-s + (0.471 − 0.881i)13-s + (−0.831 + 0.555i)14-s + (0.956 − 0.290i)15-s + (0.707 + 0.707i)16-s + (−0.634 + 0.773i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $-0.336 + 0.941i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ -0.336 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5435109522 - 0.7718104182i\)
\(L(\frac12)\) \(\approx\) \(-0.5435109522 - 0.7718104182i\)
\(L(1)\) \(\approx\) \(0.5841910041 - 0.7948053179i\)
\(L(1)\) \(\approx\) \(0.5841910041 - 0.7948053179i\)

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

−27.41135158325732176071228397263, −26.078806236527931663543435056726, −25.62901760367068835765176256185, −24.95165780418670401086004811052, −23.758701528727003948443773591843, −22.645087212172930801764543035320, −21.79845355279559303618698682460, −21.09868356345510952578465677263, −19.96911414824461302656451909433, −18.66772233000170992642096557304, −17.434884394385935381046414564047, −16.652217748291649082066659278464, −15.68263368411148859236295419923, −15.17441487510013182720274337043, −13.80089890049224582103516979089, −13.16409768185798782705501969095, −11.88644167935546520756720904613, −10.0428376320114034096175633037, −9.10828738919634324514520750731, −8.81293483221606614179695861991, −7.11571468414267007169943557485, −5.84852654084053738334663563542, −4.916771680043656327425158661167, −3.97186566192984706343160481224, −2.32925645514180319673120335203, 0.27758290109487275252878505005, 1.686632011331462706917521445045, 2.93832019417416860664832610571, 3.69218559840666334354404915939, 5.79880192819521124269155566231, 6.55664463985247067331852427135, 8.09716023767826439634989746374, 9.162661366984756977334889393242, 10.526856627683666273224146888533, 11.00387989635926644059569061335, 12.736615441607396540567652970226, 13.12355477981472056794254586701, 14.08237387894724553004237052852, 14.871089530173058313435456737708, 16.78368919887972059599467460894, 17.90773862047928063802463570246, 18.61701013809744104367981409341, 19.4312418488011033032254061479, 20.24355183047915097129923805755, 21.31317280325733803068394202922, 22.381813091112310146089414366780, 23.12563143884382700653748625706, 24.0447975397969382370255431715, 25.32977812788148654657422893852, 26.23335139382795763677863900728

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