Properties

Label 2-43e2-1.1-c1-0-127
Degree $2$
Conductor $1849$
Sign $-1$
Analytic cond. $14.7643$
Root an. cond. $3.84243$
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  + 2.17·2-s + 0.327·3-s + 2.72·4-s − 0.0263·5-s + 0.712·6-s − 3.12·7-s + 1.57·8-s − 2.89·9-s − 0.0572·10-s − 3.73·11-s + 0.893·12-s − 4.89·13-s − 6.80·14-s − 0.00862·15-s − 2.02·16-s + 5.91·17-s − 6.28·18-s + 3.96·19-s − 0.0717·20-s − 1.02·21-s − 8.12·22-s + 0.343·23-s + 0.517·24-s − 4.99·25-s − 10.6·26-s − 1.93·27-s − 8.53·28-s + ⋯
L(s)  = 1  + 1.53·2-s + 0.189·3-s + 1.36·4-s − 0.0117·5-s + 0.290·6-s − 1.18·7-s + 0.558·8-s − 0.964·9-s − 0.0180·10-s − 1.12·11-s + 0.257·12-s − 1.35·13-s − 1.81·14-s − 0.00222·15-s − 0.505·16-s + 1.43·17-s − 1.48·18-s + 0.910·19-s − 0.0160·20-s − 0.223·21-s − 1.73·22-s + 0.0717·23-s + 0.105·24-s − 0.999·25-s − 2.08·26-s − 0.371·27-s − 1.61·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-1$
Analytic conductor: \(14.7643\)
Root analytic conductor: \(3.84243\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1849,\ (\ :1/2),\ -1)\)

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)$
bad43 \( 1 \)
good2 \( 1 - 2.17T + 2T^{2} \)
3 \( 1 - 0.327T + 3T^{2} \)
5 \( 1 + 0.0263T + 5T^{2} \)
7 \( 1 + 3.12T + 7T^{2} \)
11 \( 1 + 3.73T + 11T^{2} \)
13 \( 1 + 4.89T + 13T^{2} \)
17 \( 1 - 5.91T + 17T^{2} \)
19 \( 1 - 3.96T + 19T^{2} \)
23 \( 1 - 0.343T + 23T^{2} \)
29 \( 1 + 1.32T + 29T^{2} \)
31 \( 1 - 2.16T + 31T^{2} \)
37 \( 1 + 10.0T + 37T^{2} \)
41 \( 1 - 1.94T + 41T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 5.81T + 53T^{2} \)
59 \( 1 - 0.172T + 59T^{2} \)
61 \( 1 - 5.56T + 61T^{2} \)
67 \( 1 + 1.10T + 67T^{2} \)
71 \( 1 + 5.29T + 71T^{2} \)
73 \( 1 + 7.39T + 73T^{2} \)
79 \( 1 + 5.41T + 79T^{2} \)
83 \( 1 - 2.03T + 83T^{2} \)
89 \( 1 + 4.00T + 89T^{2} \)
97 \( 1 - 9.00T + 97T^{2} \)
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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

−8.913780828900822921832121398992, −7.70372980288068402688958249164, −7.20867779718419420653638917577, −6.03953132592830415673526151996, −5.53467886855365002999635661111, −4.92292682543346583919360458787, −3.62552174163566629341856843054, −3.05617982242874895623942504645, −2.39947443490716328098582381483, 0, 2.39947443490716328098582381483, 3.05617982242874895623942504645, 3.62552174163566629341856843054, 4.92292682543346583919360458787, 5.53467886855365002999635661111, 6.03953132592830415673526151996, 7.20867779718419420653638917577, 7.70372980288068402688958249164, 8.913780828900822921832121398992

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