Properties

Label 1-133-133.132-r0-0-0
Degree $1$
Conductor $133$
Sign $1$
Analytic cond. $0.617649$
Root an. cond. $0.617649$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 15-s + 16-s − 17-s − 18-s − 20-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s − 29-s + 30-s + 31-s − 32-s + 33-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 15-s + 16-s − 17-s − 18-s − 20-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s − 29-s + 30-s + 31-s − 32-s + 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $1$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{133} (132, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9084590317\)
\(L(\frac12)\) \(\approx\) \(0.9084590317\)
\(L(1)\) \(\approx\) \(0.8936883087\)
\(L(1)\) \(\approx\) \(0.8936883087\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 + T \)
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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

−28.24758837664794843697727615596, −27.46824532055970711680240660253, −26.66137132762555261058646946522, −25.873482625399781992545440174899, −24.77625639764276167769021135215, −24.11795044361899509082713214847, −22.65193510260904049207325830085, −21.08112790359596772335582481385, −20.28573769920505928756350109642, −19.39531025590113440579401678424, −18.846640952188759442978194100199, −17.56045729012685395588945963231, −16.19422064040168239364255689780, −15.45340298751968508614649748391, −14.52780433872581024357404687792, −13.00238459337027161207626894526, −11.671390658087327129590431380805, −10.72220816155185961236418829983, −9.18601306679871263617429376743, −8.622107034578946389255971755361, −7.50667577710357321372913226853, −6.537290780686861531090829493660, −4.151323040637174249382431018531, −3.02676951049389218311006396684, −1.39047300131328585995249715062, 1.39047300131328585995249715062, 3.02676951049389218311006396684, 4.151323040637174249382431018531, 6.537290780686861531090829493660, 7.50667577710357321372913226853, 8.622107034578946389255971755361, 9.18601306679871263617429376743, 10.72220816155185961236418829983, 11.671390658087327129590431380805, 13.00238459337027161207626894526, 14.52780433872581024357404687792, 15.45340298751968508614649748391, 16.19422064040168239364255689780, 17.56045729012685395588945963231, 18.846640952188759442978194100199, 19.39531025590113440579401678424, 20.28573769920505928756350109642, 21.08112790359596772335582481385, 22.65193510260904049207325830085, 24.11795044361899509082713214847, 24.77625639764276167769021135215, 25.873482625399781992545440174899, 26.66137132762555261058646946522, 27.46824532055970711680240660253, 28.24758837664794843697727615596

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