Properties

Label 2-43e2-1.1-c1-0-10
Degree $2$
Conductor $1849$
Sign $1$
Analytic cond. $14.7643$
Root an. cond. $3.84243$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s + 5-s + 6-s − 3·7-s + 3·8-s − 2·9-s − 10-s + 12-s − 5·13-s + 3·14-s − 15-s − 16-s + 3·17-s + 2·18-s − 19-s − 20-s + 3·21-s − 7·23-s − 3·24-s − 4·25-s + 5·26-s + 5·27-s + 3·28-s − 3·29-s + 30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-s + 1.06·8-s − 2/3·9-s − 0.316·10-s + 0.288·12-s − 1.38·13-s + 0.801·14-s − 0.258·15-s − 1/4·16-s + 0.727·17-s + 0.471·18-s − 0.229·19-s − 0.223·20-s + 0.654·21-s − 1.45·23-s − 0.612·24-s − 4/5·25-s + 0.980·26-s + 0.962·27-s + 0.566·28-s − 0.557·29-s + 0.182·30-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1849,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3336119298\)
\(L(\frac12)\) \(\approx\) \(0.3336119298\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−9.619159184248389210790268829180, −8.465848922214887729446516878993, −7.85649939029523801473917002587, −6.86535113552977580278725128719, −6.02291781357434847509828575937, −5.35497377421466880095336728668, −4.40336019829144899822949439307, −3.26026786719576192842823709606, −2.08270609090444390409348198124, −0.42854195730248636535135739890, 0.42854195730248636535135739890, 2.08270609090444390409348198124, 3.26026786719576192842823709606, 4.40336019829144899822949439307, 5.35497377421466880095336728668, 6.02291781357434847509828575937, 6.86535113552977580278725128719, 7.85649939029523801473917002587, 8.465848922214887729446516878993, 9.619159184248389210790268829180

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