Properties

Label 1-193-193.76-r1-0-0
Degree $1$
Conductor $193$
Sign $-0.653 + 0.756i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(193\)
Sign: $-0.653 + 0.756i$
Analytic conductor: \(20.7407\)
Root analytic conductor: \(20.7407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{193} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 193,\ (1:\ ),\ -0.653 + 0.756i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4539526443 + 0.9915560679i\)
\(L(\frac12)\) \(\approx\) \(0.4539526443 + 0.9915560679i\)
\(L(1)\) \(\approx\) \(0.9037505052 + 0.2167164627i\)
\(L(1)\) \(\approx\) \(0.9037505052 + 0.2167164627i\)

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

−26.25257201293400311112340890720, −25.15662773131633709963436976787, −24.830062780741434512792596573497, −23.978514837378943436443201892854, −22.84975374630996171561759287520, −22.09606499224561535416413697375, −20.35306196550486250233716985764, −19.672142371676825060365001744187, −18.63321568773363200395507114511, −17.376791656848574457149102498844, −17.14501245815985441774935581817, −15.90127883902868931662834650400, −14.52808277712098938569913448810, −13.70561714398951782018652720928, −13.1316703970496371022984977885, −11.974717541888308403313943056203, −9.9008057872847307169648687944, −9.319979107039724529013564807382, −8.1442245924282605417024249489, −7.056444778526181448255918425477, −6.302305232570523599566780352008, −5.124470413546158854814558044293, −3.533745771424394097377198737078, −1.605073533273761395594055812583, −0.3764996572907518088108919320, 2.06556624581338804680801795854, 2.895568464979060990059147025620, 4.06532958185423639622233643106, 5.32914778702761055526393085019, 6.77589728103109603453584049252, 8.689502744604720414836154504126, 9.39189781931645393981821546727, 10.06637538112580472976774156947, 11.12304573064605990960423546878, 12.18985855509177528947353029575, 13.48030980963872153004672817053, 14.35986712435015109439994141402, 15.27675394468493643669657266504, 16.760915722553739799271452776210, 17.52425830742024556477392450801, 18.82481278208798606197016067284, 19.53920554243228881385646098663, 20.47267888075086833613902306652, 21.62387170023965593012738522661, 22.23543963845352827828834303151, 22.46755878724870101455114787912, 24.53556416881634493111252940113, 25.69129675829949731543985084271, 26.36703245883222617717784009335, 27.07226068145833695821614620032

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