Properties

Label 2-43e2-1.1-c3-0-102
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.40·2-s − 7.71·3-s − 6.03·4-s − 16.2·5-s − 10.8·6-s − 16.8·7-s − 19.6·8-s + 32.4·9-s − 22.8·10-s + 16.6·11-s + 46.5·12-s − 80.3·13-s − 23.5·14-s + 125.·15-s + 20.7·16-s − 86.4·17-s + 45.4·18-s − 134.·19-s + 98.3·20-s + 129.·21-s + 23.3·22-s + 92.8·23-s + 151.·24-s + 140.·25-s − 112.·26-s − 42.1·27-s + 101.·28-s + ⋯
L(s)  = 1  + 0.495·2-s − 1.48·3-s − 0.754·4-s − 1.45·5-s − 0.735·6-s − 0.908·7-s − 0.869·8-s + 1.20·9-s − 0.721·10-s + 0.455·11-s + 1.11·12-s − 1.71·13-s − 0.450·14-s + 2.16·15-s + 0.323·16-s − 1.23·17-s + 0.595·18-s − 1.62·19-s + 1.09·20-s + 1.34·21-s + 0.225·22-s + 0.841·23-s + 1.29·24-s + 1.12·25-s − 0.849·26-s − 0.300·27-s + 0.685·28-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: $\chi_{1849} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1849,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.40T + 8T^{2} \)
3 \( 1 + 7.71T + 27T^{2} \)
5 \( 1 + 16.2T + 125T^{2} \)
7 \( 1 + 16.8T + 343T^{2} \)
11 \( 1 - 16.6T + 1.33e3T^{2} \)
13 \( 1 + 80.3T + 2.19e3T^{2} \)
17 \( 1 + 86.4T + 4.91e3T^{2} \)
19 \( 1 + 134.T + 6.85e3T^{2} \)
23 \( 1 - 92.8T + 1.21e4T^{2} \)
29 \( 1 + 49.5T + 2.43e4T^{2} \)
31 \( 1 + 59.7T + 2.97e4T^{2} \)
37 \( 1 - 128.T + 5.06e4T^{2} \)
41 \( 1 - 244.T + 6.89e4T^{2} \)
47 \( 1 - 77.7T + 1.03e5T^{2} \)
53 \( 1 - 444.T + 1.48e5T^{2} \)
59 \( 1 - 539.T + 2.05e5T^{2} \)
61 \( 1 + 627.T + 2.26e5T^{2} \)
67 \( 1 - 546.T + 3.00e5T^{2} \)
71 \( 1 + 226.T + 3.57e5T^{2} \)
73 \( 1 + 736.T + 3.89e5T^{2} \)
79 \( 1 - 440.T + 4.93e5T^{2} \)
83 \( 1 + 85.4T + 5.71e5T^{2} \)
89 \( 1 + 882.T + 7.04e5T^{2} \)
97 \( 1 + 6.99T + 9.12e5T^{2} \)
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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

−8.579462399673829341817209178537, −7.41014610129913320250947565646, −6.76021308478540355197860668211, −6.06059309747770070069587664110, −5.03472353157621637063580892214, −4.41906437980612740377923416196, −3.91452114786444468217752407137, −2.65118363563922265588572094238, −0.52552492664752948945997647760, 0, 0.52552492664752948945997647760, 2.65118363563922265588572094238, 3.91452114786444468217752407137, 4.41906437980612740377923416196, 5.03472353157621637063580892214, 6.06059309747770070069587664110, 6.76021308478540355197860668211, 7.41014610129913320250947565646, 8.579462399673829341817209178537

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