Properties

Label 2-1755-1.1-c1-0-48
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618·2-s − 1.61·4-s + 5-s + 1.23·7-s + 2.23·8-s − 0.618·10-s − 3.23·11-s − 13-s − 0.763·14-s + 1.85·16-s + 0.236·17-s − 5.47·19-s − 1.61·20-s + 2.00·22-s + 1.47·23-s + 25-s + 0.618·26-s − 2.00·28-s + 5.23·29-s − 3·31-s − 5.61·32-s − 0.145·34-s + 1.23·35-s − 6·37-s + 3.38·38-s + 2.23·40-s − 10.4·41-s + ⋯
L(s)  = 1  − 0.437·2-s − 0.809·4-s + 0.447·5-s + 0.467·7-s + 0.790·8-s − 0.195·10-s − 0.975·11-s − 0.277·13-s − 0.204·14-s + 0.463·16-s + 0.0572·17-s − 1.25·19-s − 0.361·20-s + 0.426·22-s + 0.306·23-s + 0.200·25-s + 0.121·26-s − 0.377·28-s + 0.972·29-s − 0.538·31-s − 0.993·32-s − 0.0250·34-s + 0.208·35-s − 0.986·37-s + 0.548·38-s + 0.353·40-s − 1.63·41-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: \(1\)
Selberg data: \((2,\ 1755,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.618T + 2T^{2} \)
7 \( 1 - 1.23T + 7T^{2} \)
11 \( 1 + 3.23T + 11T^{2} \)
17 \( 1 - 0.236T + 17T^{2} \)
19 \( 1 + 5.47T + 19T^{2} \)
23 \( 1 - 1.47T + 23T^{2} \)
29 \( 1 - 5.23T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 - 6.47T + 43T^{2} \)
47 \( 1 - 2.47T + 47T^{2} \)
53 \( 1 - 6.70T + 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 11T + 61T^{2} \)
67 \( 1 + 11.7T + 67T^{2} \)
71 \( 1 - 8.76T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 6.70T + 79T^{2} \)
83 \( 1 - 8.23T + 83T^{2} \)
89 \( 1 + 9.70T + 89T^{2} \)
97 \( 1 + 6.47T + 97T^{2} \)
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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

−8.811131808105939576959172104322, −8.311393525815262462019885310877, −7.53387510199655712037428178449, −6.58740070013619464740533919935, −5.43518177573231588486243515645, −4.91008018531659278825814255806, −3.99269120246150146571910473633, −2.67922708913879507202262590878, −1.53635149057595003695857915883, 0, 1.53635149057595003695857915883, 2.67922708913879507202262590878, 3.99269120246150146571910473633, 4.91008018531659278825814255806, 5.43518177573231588486243515645, 6.58740070013619464740533919935, 7.53387510199655712037428178449, 8.311393525815262462019885310877, 8.811131808105939576959172104322

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