Properties

Label 2-1755-1.1-c1-0-27
Degree $2$
Conductor $1755$
Sign $1$
Analytic cond. $14.0137$
Root an. cond. $3.74349$
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  + 0.401·2-s − 1.83·4-s + 5-s + 2.55·7-s − 1.54·8-s + 0.401·10-s + 6.34·11-s + 13-s + 1.02·14-s + 3.05·16-s − 1.45·17-s − 2.89·19-s − 1.83·20-s + 2.55·22-s − 1.87·23-s + 25-s + 0.401·26-s − 4.70·28-s − 3.57·29-s − 5.67·31-s + 4.31·32-s − 0.583·34-s + 2.55·35-s + 6.18·37-s − 1.16·38-s − 1.54·40-s + 3.33·41-s + ⋯
L(s)  = 1  + 0.284·2-s − 0.919·4-s + 0.447·5-s + 0.966·7-s − 0.545·8-s + 0.127·10-s + 1.91·11-s + 0.277·13-s + 0.274·14-s + 0.764·16-s − 0.352·17-s − 0.664·19-s − 0.411·20-s + 0.543·22-s − 0.391·23-s + 0.200·25-s + 0.0787·26-s − 0.888·28-s − 0.664·29-s − 1.01·31-s + 0.762·32-s − 0.100·34-s + 0.432·35-s + 1.01·37-s − 0.188·38-s − 0.243·40-s + 0.520·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(14.0137\)
Root analytic conductor: \(3.74349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1755,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.115314330\)
\(L(\frac12)\) \(\approx\) \(2.115314330\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
13 \( 1 - T \)
good2 \( 1 - 0.401T + 2T^{2} \)
7 \( 1 - 2.55T + 7T^{2} \)
11 \( 1 - 6.34T + 11T^{2} \)
17 \( 1 + 1.45T + 17T^{2} \)
19 \( 1 + 2.89T + 19T^{2} \)
23 \( 1 + 1.87T + 23T^{2} \)
29 \( 1 + 3.57T + 29T^{2} \)
31 \( 1 + 5.67T + 31T^{2} \)
37 \( 1 - 6.18T + 37T^{2} \)
41 \( 1 - 3.33T + 41T^{2} \)
43 \( 1 - 9.46T + 43T^{2} \)
47 \( 1 - 2.77T + 47T^{2} \)
53 \( 1 + 9.34T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 - 7.83T + 61T^{2} \)
67 \( 1 - 9.22T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 - 14.8T + 73T^{2} \)
79 \( 1 - 4.01T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + 8.56T + 89T^{2} \)
97 \( 1 + 7.57T + 97T^{2} \)
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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

−9.279404073425567793687906286370, −8.649522857896023912503246904432, −7.900442446367974860357339100347, −6.74857319968620920823707497689, −5.99124151374311952954619065021, −5.17907179479721125144000335796, −4.17870759232131953549163500660, −3.80188584821287086091297002536, −2.14891834200856410649977762096, −1.04596833760551921228142955261, 1.04596833760551921228142955261, 2.14891834200856410649977762096, 3.80188584821287086091297002536, 4.17870759232131953549163500660, 5.17907179479721125144000335796, 5.99124151374311952954619065021, 6.74857319968620920823707497689, 7.900442446367974860357339100347, 8.649522857896023912503246904432, 9.279404073425567793687906286370

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