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Properties

Label 1-12-12.11-r0-0-0
Degree $1$
Conductor $12$
Sign $1$
Analytic cond. $0.0557277$
Root an. cond. $0.0557277$
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  − 5-s − 7-s + 11-s + 13-s − 17-s − 19-s + 23-s + 25-s − 29-s − 31-s + 35-s + 37-s − 41-s − 43-s + 47-s + 49-s − 53-s − 55-s + 59-s + 61-s − 65-s − 67-s + 71-s + 73-s − 77-s − 79-s + 83-s + ⋯
L(s)  = 1  − 5-s − 7-s + 11-s + 13-s − 17-s − 19-s + 23-s + 25-s − 29-s − 31-s + 35-s + 37-s − 41-s − 43-s + 47-s + 49-s − 53-s − 55-s + 59-s + 61-s − 65-s − 67-s + 71-s + 73-s − 77-s − 79-s + 83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $1$
Analytic conductor: \(0.0557277\)
Root analytic conductor: \(0.0557277\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 12,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4985570024\)
\(L(\frac12)\) \(\approx\) \(0.4985570024\)
\(L(1)\) \(\approx\) \(0.7603459963\)
\(L(1)\) \(\approx\) \(0.7603459963\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 - T \)
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

−44.94882250215854377900265729379, −43.192847102617221657447923692506, −42.23514301791452202974955316843, −40.48415475080402043184282320936, −38.97399882227494028953955236153, −37.92681682137521283528523170529, −35.77804457652351027241685311662, −35.02737848501270258165386682011, −33.037132879506503601921758064941, −31.648077614940686366898633770299, −30.20400655643888185431053894859, −28.44220325774560811850882430583, −27.013943985850672344345415796374, −25.411633892392695962033821338368, −23.561319713137844716347932732028, −22.285839107226887658739733010514, −20.10392819124581857313303938476, −18.884369457120650808166588937461, −16.632633274523762123171390835505, −15.181480875888216735397832429452, −12.966178808028845374957099032698, −11.18839274507490915416358679559, −8.890592958726741509335625537170, −6.69222332050013115991012819444, −3.80462763305086509714431754003, 3.80462763305086509714431754003, 6.69222332050013115991012819444, 8.890592958726741509335625537170, 11.18839274507490915416358679559, 12.966178808028845374957099032698, 15.181480875888216735397832429452, 16.632633274523762123171390835505, 18.884369457120650808166588937461, 20.10392819124581857313303938476, 22.285839107226887658739733010514, 23.561319713137844716347932732028, 25.411633892392695962033821338368, 27.013943985850672344345415796374, 28.44220325774560811850882430583, 30.20400655643888185431053894859, 31.648077614940686366898633770299, 33.037132879506503601921758064941, 35.02737848501270258165386682011, 35.77804457652351027241685311662, 37.92681682137521283528523170529, 38.97399882227494028953955236153, 40.48415475080402043184282320936, 42.23514301791452202974955316843, 43.192847102617221657447923692506, 44.94882250215854377900265729379

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