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Properties

Label 1-181-181.13-r0-0-0
Degree $1$
Conductor $181$
Sign $0.765 - 0.643i$
Analytic cond. $0.840560$
Root an. cond. $0.840560$
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  + (0.848 − 0.529i)2-s + (0.990 + 0.139i)3-s + (0.438 − 0.898i)4-s + (0.669 + 0.743i)5-s + (0.913 − 0.406i)6-s + (−0.5 − 0.866i)7-s + (−0.104 − 0.994i)8-s + (0.961 + 0.275i)9-s + (0.961 + 0.275i)10-s + (−0.997 + 0.0697i)11-s + (0.559 − 0.829i)12-s + (−0.882 + 0.469i)13-s + (−0.882 − 0.469i)14-s + (0.559 + 0.829i)15-s + (−0.615 − 0.788i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.848 − 0.529i)2-s + (0.990 + 0.139i)3-s + (0.438 − 0.898i)4-s + (0.669 + 0.743i)5-s + (0.913 − 0.406i)6-s + (−0.5 − 0.866i)7-s + (−0.104 − 0.994i)8-s + (0.961 + 0.275i)9-s + (0.961 + 0.275i)10-s + (−0.997 + 0.0697i)11-s + (0.559 − 0.829i)12-s + (−0.882 + 0.469i)13-s + (−0.882 − 0.469i)14-s + (0.559 + 0.829i)15-s + (−0.615 − 0.788i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(181\)
Sign: $0.765 - 0.643i$
Analytic conductor: \(0.840560\)
Root analytic conductor: \(0.840560\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{181} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 181,\ (0:\ ),\ 0.765 - 0.643i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.281057817 - 0.8314472516i\)
\(L(\frac12)\) \(\approx\) \(2.281057817 - 0.8314472516i\)
\(L(1)\) \(\approx\) \(2.020414947 - 0.5396387736i\)
\(L(1)\) \(\approx\) \(2.020414947 - 0.5396387736i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad181 \( 1 \)
good2 \( 1 + (0.848 - 0.529i)T \)
3 \( 1 + (0.990 + 0.139i)T \)
5 \( 1 + (0.669 + 0.743i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.997 + 0.0697i)T \)
13 \( 1 + (-0.882 + 0.469i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.241 + 0.970i)T \)
29 \( 1 + (-0.104 - 0.994i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.374 - 0.927i)T \)
41 \( 1 + (-0.719 - 0.694i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.559 - 0.829i)T \)
53 \( 1 + (-0.719 + 0.694i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.913 - 0.406i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (-0.719 + 0.694i)T \)
83 \( 1 + (0.0348 + 0.999i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (-0.997 - 0.0697i)T \)
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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

−27.03839731291720936811808416764, −26.033167874292368710259774314293, −25.35937274648731007605713338603, −24.47220364740301628044314619204, −24.10755651954347916181789729598, −22.36672111514250382892592445128, −21.791252145666486426061492689353, −20.61307451983408242677836283987, −20.187531690953451192135109403220, −18.62735337216651646151748923540, −17.63817639276144654985211798463, −16.22744179738958806404421578502, −15.570408466416655124039548703851, −14.57813944194479423358868561892, −13.45092014114668055061089614630, −12.91062121390767708251565226528, −12.030447726853769330413024738190, −10.06386670488226558013476591986, −8.94776426188890529327533406167, −8.08733513095524011920660371234, −6.861631641320280861562467398390, −5.54950814821094253109845128183, −4.65257193517536936676023184751, −2.96035472170301464552556602918, −2.264366610207475224212473971749, 1.90898044872196509243868109603, 2.86110996742363692924145094380, 3.86502351181383301605363652795, 5.16887293280234209038174325140, 6.70052473345655800177589775762, 7.51945109097912054095939397890, 9.5065305205127832939902234602, 10.08260758472045520989698615008, 11.02788127837997731940184343401, 12.63801544165523246915174363600, 13.67680512660024709509231864122, 13.954777606511865534027805767846, 15.16861766969763295796078465640, 15.96903357636229596173620396183, 17.651810211947723047946084495211, 18.89834848908348536654763125904, 19.6503752597715709471528903691, 20.52010922089117133123970731281, 21.4706714711141301432452905300, 22.15775586990301410607435040584, 23.2603647052204938170085292572, 24.29157858134917115981952702867, 25.2408217422351189747844916897, 26.392918771545353876069831753495, 26.756200368613867884050238485107

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