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Properties

Label 1-193-193.88-r1-0-0
Degree $1$
Conductor $193$
Sign $0.530 - 0.847i$
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  + (−0.980 + 0.195i)2-s + (−0.382 − 0.923i)3-s + (0.923 − 0.382i)4-s + (−0.995 + 0.0980i)5-s + (0.555 + 0.831i)6-s + (−0.707 + 0.707i)7-s + (−0.831 + 0.555i)8-s + (−0.707 + 0.707i)9-s + (0.956 − 0.290i)10-s + (−0.634 − 0.773i)11-s + (−0.707 − 0.707i)12-s + (−0.290 + 0.956i)13-s + (0.555 − 0.831i)14-s + (0.471 + 0.881i)15-s + (0.707 − 0.707i)16-s + (0.995 + 0.0980i)17-s + ⋯
L(s)  = 1  + (−0.980 + 0.195i)2-s + (−0.382 − 0.923i)3-s + (0.923 − 0.382i)4-s + (−0.995 + 0.0980i)5-s + (0.555 + 0.831i)6-s + (−0.707 + 0.707i)7-s + (−0.831 + 0.555i)8-s + (−0.707 + 0.707i)9-s + (0.956 − 0.290i)10-s + (−0.634 − 0.773i)11-s + (−0.707 − 0.707i)12-s + (−0.290 + 0.956i)13-s + (0.555 − 0.831i)14-s + (0.471 + 0.881i)15-s + (0.707 − 0.707i)16-s + (0.995 + 0.0980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $0.530 - 0.847i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ 0.530 - 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3193227931 - 0.1767381624i\)
\(L(\frac12)\) \(\approx\) \(0.3193227931 - 0.1767381624i\)
\(L(1)\) \(\approx\) \(0.4108805005 - 0.04389279971i\)
\(L(1)\) \(\approx\) \(0.4108805005 - 0.04389279971i\)

Euler product

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

−27.04208888573555418417417318565, −26.23331052924277861325243712325, −25.599499367412346557061564628320, −24.02052164955533735974536210284, −23.09141699195357345482398734852, −22.24282003582036979661234895348, −20.87343708442255033056519373492, −20.14095361852689532574642002593, −19.55006105342463072497837602513, −18.16997988744697791826265979773, −17.24460586744746766727726633739, −16.21996057440937857776666458369, −15.73856164556207311028520088199, −14.76465403773197111610198252081, −12.74371841832626539394226320927, −11.93637722958943142273065972144, −10.676361715003984888890099755677, −10.1987010814212276023352362250, −9.10032145918463404145904402591, −7.81682096940808473062498309977, −6.97696060894961772741501927556, −5.34459207399078167762322523641, −3.88178876205980337687064909768, −2.951568912413495948249558300861, −0.59633504522778459374831001044, 0.34306583053773439494841771097, 1.980075722274889601402601136693, 3.31695549698227151439173815611, 5.58108827340706516139211565301, 6.41833357664692974716673490938, 7.64539380219223666797285777318, 8.20012667173147412953370453829, 9.51187737240639051007094632323, 10.84856229707574171334404552529, 11.85872086816781231051488834321, 12.38218007756911388720217226583, 13.99317729851876492439762329357, 15.26509197577413020653934639647, 16.33112732930599956060555347046, 16.812363034092459287560444877435, 18.4618968719452700614617524950, 18.79894256243955105079018082881, 19.379357017685964218689865976263, 20.58686943673822058391545423821, 22.00701477308580693517943952414, 23.242439239932505618265959753322, 23.956907750384894784313889309419, 24.730165194989915510350352560269, 25.811456601056769332147768033153, 26.55895718769884204009323150874

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