Properties

Label 2-1573-1.1-c1-0-38
Degree $2$
Conductor $1573$
Sign $1$
Analytic cond. $12.5604$
Root an. cond. $3.54407$
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  + 0.919·2-s − 1.06·3-s − 1.15·4-s + 3.26·5-s − 0.982·6-s + 2.12·7-s − 2.90·8-s − 1.85·9-s + 3.00·10-s + 1.23·12-s + 13-s + 1.95·14-s − 3.49·15-s − 0.358·16-s + 2.81·17-s − 1.70·18-s + 0.663·19-s − 3.77·20-s − 2.26·21-s + 4.97·23-s + 3.09·24-s + 5.69·25-s + 0.919·26-s + 5.19·27-s − 2.45·28-s − 4.81·29-s − 3.21·30-s + ⋯
L(s)  = 1  + 0.650·2-s − 0.616·3-s − 0.577·4-s + 1.46·5-s − 0.401·6-s + 0.803·7-s − 1.02·8-s − 0.619·9-s + 0.950·10-s + 0.356·12-s + 0.277·13-s + 0.522·14-s − 0.901·15-s − 0.0895·16-s + 0.682·17-s − 0.402·18-s + 0.152·19-s − 0.844·20-s − 0.495·21-s + 1.03·23-s + 0.632·24-s + 1.13·25-s + 0.180·26-s + 0.998·27-s − 0.463·28-s − 0.893·29-s − 0.586·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1573\)    =    \(11^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(12.5604\)
Root analytic conductor: \(3.54407\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1573,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.169116691\)
\(L(\frac12)\) \(\approx\) \(2.169116691\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 - T \)
good2 \( 1 - 0.919T + 2T^{2} \)
3 \( 1 + 1.06T + 3T^{2} \)
5 \( 1 - 3.26T + 5T^{2} \)
7 \( 1 - 2.12T + 7T^{2} \)
17 \( 1 - 2.81T + 17T^{2} \)
19 \( 1 - 0.663T + 19T^{2} \)
23 \( 1 - 4.97T + 23T^{2} \)
29 \( 1 + 4.81T + 29T^{2} \)
31 \( 1 - 1.19T + 31T^{2} \)
37 \( 1 - 4.62T + 37T^{2} \)
41 \( 1 - 7.53T + 41T^{2} \)
43 \( 1 + 0.661T + 43T^{2} \)
47 \( 1 + 7.67T + 47T^{2} \)
53 \( 1 + 8.04T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 - 7.05T + 61T^{2} \)
67 \( 1 - 14.7T + 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 + 1.38T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + 3.53T + 83T^{2} \)
89 \( 1 - 8.08T + 89T^{2} \)
97 \( 1 - 7.11T + 97T^{2} \)
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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

−9.484318014416519881296046989521, −8.758117660479301238440455234979, −7.926815049458508933626559925767, −6.58139749078352881688395760485, −5.88250774530348536779742734793, −5.30016004562274373804127981766, −4.80992428139957149452950222368, −3.49326561385910212256933873209, −2.39537679345129174628231441811, −1.02977801802582399457253486274, 1.02977801802582399457253486274, 2.39537679345129174628231441811, 3.49326561385910212256933873209, 4.80992428139957149452950222368, 5.30016004562274373804127981766, 5.88250774530348536779742734793, 6.58139749078352881688395760485, 7.926815049458508933626559925767, 8.758117660479301238440455234979, 9.484318014416519881296046989521

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