Properties

Label 2-43e2-1.1-c1-0-67
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.97·2-s − 0.955·3-s + 1.90·4-s + 3.57·5-s − 1.88·6-s + 2.56·7-s − 0.191·8-s − 2.08·9-s + 7.06·10-s − 2.18·11-s − 1.81·12-s + 1.23·13-s + 5.07·14-s − 3.41·15-s − 4.18·16-s + 2.68·17-s − 4.12·18-s + 8.30·19-s + 6.80·20-s − 2.45·21-s − 4.32·22-s + 1.65·23-s + 0.183·24-s + 7.79·25-s + 2.43·26-s + 4.86·27-s + 4.88·28-s + ⋯
L(s)  = 1  + 1.39·2-s − 0.551·3-s + 0.951·4-s + 1.59·5-s − 0.770·6-s + 0.970·7-s − 0.0677·8-s − 0.695·9-s + 2.23·10-s − 0.659·11-s − 0.525·12-s + 0.341·13-s + 1.35·14-s − 0.882·15-s − 1.04·16-s + 0.650·17-s − 0.971·18-s + 1.90·19-s + 1.52·20-s − 0.535·21-s − 0.921·22-s + 0.344·23-s + 0.0373·24-s + 1.55·25-s + 0.477·26-s + 0.935·27-s + 0.923·28-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: \(0\)
Selberg data: \((2,\ 1849,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.185694960\)
\(L(\frac12)\) \(\approx\) \(4.185694960\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.97T + 2T^{2} \)
3 \( 1 + 0.955T + 3T^{2} \)
5 \( 1 - 3.57T + 5T^{2} \)
7 \( 1 - 2.56T + 7T^{2} \)
11 \( 1 + 2.18T + 11T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 - 2.68T + 17T^{2} \)
19 \( 1 - 8.30T + 19T^{2} \)
23 \( 1 - 1.65T + 23T^{2} \)
29 \( 1 - 3.71T + 29T^{2} \)
31 \( 1 + 4.21T + 31T^{2} \)
37 \( 1 - 9.61T + 37T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
47 \( 1 - 1.54T + 47T^{2} \)
53 \( 1 - 3.99T + 53T^{2} \)
59 \( 1 - 4.71T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + 2.17T + 67T^{2} \)
71 \( 1 + 2.35T + 71T^{2} \)
73 \( 1 + 5.37T + 73T^{2} \)
79 \( 1 + 1.60T + 79T^{2} \)
83 \( 1 - 7.02T + 83T^{2} \)
89 \( 1 + 9.42T + 89T^{2} \)
97 \( 1 + 6.48T + 97T^{2} \)
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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

−9.353644461450891209461783638433, −8.478031081361705536252259657019, −7.41343125094018982170407548548, −6.35789852077102338557358745448, −5.66715085457959336377021274546, −5.30290068547691390319174592682, −4.75627429187205668327134640796, −3.27696391298969689795978066729, −2.54805523161992435098731333632, −1.31598956612876023637526308771, 1.31598956612876023637526308771, 2.54805523161992435098731333632, 3.27696391298969689795978066729, 4.75627429187205668327134640796, 5.30290068547691390319174592682, 5.66715085457959336377021274546, 6.35789852077102338557358745448, 7.41343125094018982170407548548, 8.478031081361705536252259657019, 9.353644461450891209461783638433

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