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Properties

Label 1-181-181.9-r0-0-0
Degree $1$
Conductor $181$
Sign $0.413 - 0.910i$
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  + (−0.719 − 0.694i)2-s + (0.559 − 0.829i)3-s + (0.0348 + 0.999i)4-s + (0.913 + 0.406i)5-s + (−0.978 + 0.207i)6-s + (−0.5 + 0.866i)7-s + (0.669 − 0.743i)8-s + (−0.374 − 0.927i)9-s + (−0.374 − 0.927i)10-s + (−0.882 − 0.469i)11-s + (0.848 + 0.529i)12-s + (0.961 + 0.275i)13-s + (0.961 − 0.275i)14-s + (0.848 − 0.529i)15-s + (−0.997 + 0.0697i)16-s + (0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (−0.719 − 0.694i)2-s + (0.559 − 0.829i)3-s + (0.0348 + 0.999i)4-s + (0.913 + 0.406i)5-s + (−0.978 + 0.207i)6-s + (−0.5 + 0.866i)7-s + (0.669 − 0.743i)8-s + (−0.374 − 0.927i)9-s + (−0.374 − 0.927i)10-s + (−0.882 − 0.469i)11-s + (0.848 + 0.529i)12-s + (0.961 + 0.275i)13-s + (0.961 − 0.275i)14-s + (0.848 − 0.529i)15-s + (−0.997 + 0.0697i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(181\)
Sign: $0.413 - 0.910i$
Analytic conductor: \(0.840560\)
Root analytic conductor: \(0.840560\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{181} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 181,\ (0:\ ),\ 0.413 - 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9034071239 - 0.5820036714i\)
\(L(\frac12)\) \(\approx\) \(0.9034071239 - 0.5820036714i\)
\(L(1)\) \(\approx\) \(0.9068386579 - 0.4083695680i\)
\(L(1)\) \(\approx\) \(0.9068386579 - 0.4083695680i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad181 \( 1 \)
good2 \( 1 + (-0.719 - 0.694i)T \)
3 \( 1 + (0.559 - 0.829i)T \)
5 \( 1 + (0.913 + 0.406i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.882 - 0.469i)T \)
13 \( 1 + (0.961 + 0.275i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + T \)
23 \( 1 + (0.990 - 0.139i)T \)
29 \( 1 + (0.669 - 0.743i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.438 + 0.898i)T \)
41 \( 1 + (-0.615 - 0.788i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.848 + 0.529i)T \)
53 \( 1 + (-0.615 + 0.788i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (-0.978 + 0.207i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (-0.615 + 0.788i)T \)
83 \( 1 + (-0.241 + 0.970i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (-0.882 + 0.469i)T \)
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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

−27.206978416816028806560096390164, −26.40153781338507834774488373750, −25.54904210074863424031587920001, −25.229133830504451487162714465251, −23.70360428945984642116162765603, −22.917121989994251780449812886951, −21.51073819582176760002171494427, −20.50570501697011636747643306532, −19.94737907661084484513601064652, −18.59930116358777455284111278009, −17.58246686847778401258791216003, −16.53819530012119563559978231687, −16.02062484713803137042763473678, −14.83730462172134823095213265211, −13.837143703058106736439931682729, −13.06550560787154357775821141873, −10.74270591088083338547324047410, −10.19443865569983979444564777525, −9.33066581347210051066468318315, −8.333490727685403023258597730784, −7.202600571329024659952208995521, −5.74942436053815078465535315824, −4.85694920105963095381497463479, −3.201098841979879169515723690887, −1.44358332945374817738282352780, 1.293316079565837715131674581495, 2.64848606755841477156934909442, 3.17655267857080002062168252651, 5.63989106716022986788987700811, 6.78926108355574744130976633214, 7.98725345219828355679475589914, 9.037948424882350741778744495515, 9.77248000780475848167880021390, 11.11478054819864398529818953537, 12.19687446772873189261597013243, 13.26221344587809467980719033211, 13.82235954721969239938195516007, 15.414522520792093429682272816754, 16.6283631450351699135288935076, 17.92785266387712907419237694207, 18.67550952288333016277564476866, 18.869824091718527017910136010217, 20.4442590581001457637882912348, 21.06613289316663257606906951836, 22.06798098276727655178281215052, 23.22371306594479847931156028507, 24.750402242201199069260658839508, 25.44037660084862308632969566076, 26.02741851616085219794481825247, 26.96730102643366855697870755794

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