Properties

Label 2-43e2-1.1-c3-0-196
Degree $2$
Conductor $1849$
Sign $1$
Analytic cond. $109.094$
Root an. cond. $10.4448$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.95·2-s − 0.318·3-s + 16.5·4-s + 12.4·5-s + 1.57·6-s + 3.53·7-s − 42.2·8-s − 26.8·9-s − 61.5·10-s + 9.73·11-s − 5.26·12-s + 58.2·13-s − 17.5·14-s − 3.95·15-s + 77.1·16-s + 96.5·17-s + 133.·18-s + 130.·19-s + 205.·20-s − 1.12·21-s − 48.2·22-s + 86.1·23-s + 13.4·24-s + 29.2·25-s − 288.·26-s + 17.1·27-s + 58.4·28-s + ⋯
L(s)  = 1  − 1.75·2-s − 0.0612·3-s + 2.06·4-s + 1.11·5-s + 0.107·6-s + 0.190·7-s − 1.86·8-s − 0.996·9-s − 1.94·10-s + 0.266·11-s − 0.126·12-s + 1.24·13-s − 0.334·14-s − 0.0680·15-s + 1.20·16-s + 1.37·17-s + 1.74·18-s + 1.57·19-s + 2.29·20-s − 0.0117·21-s − 0.467·22-s + 0.781·23-s + 0.114·24-s + 0.233·25-s − 2.17·26-s + 0.122·27-s + 0.394·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $1$
Analytic conductor: \(109.094\)
Root analytic conductor: \(10.4448\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1849,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.480905803\)
\(L(\frac12)\) \(\approx\) \(1.480905803\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + 4.95T + 8T^{2} \)
3 \( 1 + 0.318T + 27T^{2} \)
5 \( 1 - 12.4T + 125T^{2} \)
7 \( 1 - 3.53T + 343T^{2} \)
11 \( 1 - 9.73T + 1.33e3T^{2} \)
13 \( 1 - 58.2T + 2.19e3T^{2} \)
17 \( 1 - 96.5T + 4.91e3T^{2} \)
19 \( 1 - 130.T + 6.85e3T^{2} \)
23 \( 1 - 86.1T + 1.21e4T^{2} \)
29 \( 1 - 78.0T + 2.43e4T^{2} \)
31 \( 1 + 103.T + 2.97e4T^{2} \)
37 \( 1 - 433.T + 5.06e4T^{2} \)
41 \( 1 + 265.T + 6.89e4T^{2} \)
47 \( 1 - 515.T + 1.03e5T^{2} \)
53 \( 1 + 142.T + 1.48e5T^{2} \)
59 \( 1 + 441.T + 2.05e5T^{2} \)
61 \( 1 - 465.T + 2.26e5T^{2} \)
67 \( 1 - 445.T + 3.00e5T^{2} \)
71 \( 1 - 978.T + 3.57e5T^{2} \)
73 \( 1 + 28.4T + 3.89e5T^{2} \)
79 \( 1 - 6.16T + 4.93e5T^{2} \)
83 \( 1 + 773.T + 5.71e5T^{2} \)
89 \( 1 + 18.0T + 7.04e5T^{2} \)
97 \( 1 - 1.46e3T + 9.12e5T^{2} \)
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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

−9.039836568011114027219761597669, −8.226705842061585110339069797052, −7.64391055688860842364221123885, −6.61319497091861550130583094164, −5.90396125752202013731673457425, −5.28154865305857254645012452083, −3.42210553818106904708050034289, −2.56166213998875734825020136261, −1.38851330507721755634906591848, −0.854683464729087682242144857370, 0.854683464729087682242144857370, 1.38851330507721755634906591848, 2.56166213998875734825020136261, 3.42210553818106904708050034289, 5.28154865305857254645012452083, 5.90396125752202013731673457425, 6.61319497091861550130583094164, 7.64391055688860842364221123885, 8.226705842061585110339069797052, 9.039836568011114027219761597669

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