Properties

Label 2-1911-1.1-c1-0-16
Degree $2$
Conductor $1911$
Sign $1$
Analytic cond. $15.2594$
Root an. cond. $3.90632$
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  − 0.583·2-s − 3-s − 1.65·4-s + 0.00312·5-s + 0.583·6-s + 2.13·8-s + 9-s − 0.00182·10-s + 4.94·11-s + 1.65·12-s + 13-s − 0.00312·15-s + 2.07·16-s − 5.87·17-s − 0.583·18-s + 5.38·19-s − 0.00519·20-s − 2.88·22-s + 4.23·23-s − 2.13·24-s − 4.99·25-s − 0.583·26-s − 27-s − 8.87·29-s + 0.00182·30-s − 5.36·31-s − 5.48·32-s + ⋯
L(s)  = 1  − 0.412·2-s − 0.577·3-s − 0.829·4-s + 0.00139·5-s + 0.238·6-s + 0.755·8-s + 0.333·9-s − 0.000577·10-s + 1.49·11-s + 0.479·12-s + 0.277·13-s − 0.000807·15-s + 0.518·16-s − 1.42·17-s − 0.137·18-s + 1.23·19-s − 0.00116·20-s − 0.615·22-s + 0.882·23-s − 0.435·24-s − 0.999·25-s − 0.114·26-s − 0.192·27-s − 1.64·29-s + 0.000333·30-s − 0.964·31-s − 0.968·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(15.2594\)
Root analytic conductor: \(3.90632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9266438076\)
\(L(\frac12)\) \(\approx\) \(0.9266438076\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 \)
13 \( 1 - T \)
good2 \( 1 + 0.583T + 2T^{2} \)
5 \( 1 - 0.00312T + 5T^{2} \)
11 \( 1 - 4.94T + 11T^{2} \)
17 \( 1 + 5.87T + 17T^{2} \)
19 \( 1 - 5.38T + 19T^{2} \)
23 \( 1 - 4.23T + 23T^{2} \)
29 \( 1 + 8.87T + 29T^{2} \)
31 \( 1 + 5.36T + 31T^{2} \)
37 \( 1 + 4.67T + 37T^{2} \)
41 \( 1 - 5.05T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + 1.95T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + 6.08T + 59T^{2} \)
61 \( 1 - 9.96T + 61T^{2} \)
67 \( 1 - 5.00T + 67T^{2} \)
71 \( 1 - 9.08T + 71T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 - 4.62T + 83T^{2} \)
89 \( 1 - 5.54T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−9.244214363606207568549191635804, −8.733128408957743009985953327039, −7.53693542884532759432696374847, −6.99571038748416685407063315905, −5.93374416084034738464002721713, −5.23293804914317714382743521090, −4.17405424024885681827573459586, −3.68083494562152204600711295298, −1.86393333106623526585966051199, −0.74324660424758292401476420494, 0.74324660424758292401476420494, 1.86393333106623526585966051199, 3.68083494562152204600711295298, 4.17405424024885681827573459586, 5.23293804914317714382743521090, 5.93374416084034738464002721713, 6.99571038748416685407063315905, 7.53693542884532759432696374847, 8.733128408957743009985953327039, 9.244214363606207568549191635804

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