Properties

Label 2-2156-1.1-c1-0-2
Degree $2$
Conductor $2156$
Sign $1$
Analytic cond. $17.2157$
Root an. cond. $4.14918$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82·3-s − 1.41·5-s + 5.00·9-s − 11-s + 1.41·13-s + 4.00·15-s − 7.07·17-s − 2.82·19-s + 4·23-s − 2.99·25-s − 5.65·27-s + 5.65·31-s + 2.82·33-s − 8·37-s − 4.00·39-s − 9.89·41-s + 4·43-s − 7.07·45-s + 20.0·51-s − 6·53-s + 1.41·55-s + 8.00·57-s − 8.48·59-s + 1.41·61-s − 2.00·65-s − 8·67-s − 11.3·69-s + ⋯
L(s)  = 1  − 1.63·3-s − 0.632·5-s + 1.66·9-s − 0.301·11-s + 0.392·13-s + 1.03·15-s − 1.71·17-s − 0.648·19-s + 0.834·23-s − 0.599·25-s − 1.08·27-s + 1.01·31-s + 0.492·33-s − 1.31·37-s − 0.640·39-s − 1.54·41-s + 0.609·43-s − 1.05·45-s + 2.80·51-s − 0.824·53-s + 0.190·55-s + 1.05·57-s − 1.10·59-s + 0.181·61-s − 0.248·65-s − 0.977·67-s − 1.36·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(17.2157\)
Root analytic conductor: \(4.14918\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2156,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4905650306\)
\(L(\frac12)\) \(\approx\) \(0.4905650306\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good3 \( 1 + 2.82T + 3T^{2} \)
5 \( 1 + 1.41T + 5T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 + 7.07T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + 9.89T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 8.48T + 59T^{2} \)
61 \( 1 - 1.41T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 - 2.82T + 83T^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 + 9.89T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.036499773228396454015948752073, −8.293465264685558393337617789661, −7.28301092137744850211042912656, −6.56791064388057944651969577390, −6.06771649936283120742517768043, −4.93522728345461897691648231740, −4.56479561533355029929680724107, −3.47239683474665217254217736922, −1.95072378154726744085075568386, −0.48581176627780197504972396988, 0.48581176627780197504972396988, 1.95072378154726744085075568386, 3.47239683474665217254217736922, 4.56479561533355029929680724107, 4.93522728345461897691648231740, 6.06771649936283120742517768043, 6.56791064388057944651969577390, 7.28301092137744850211042912656, 8.293465264685558393337617789661, 9.036499773228396454015948752073

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