Properties

Label 2-1856-1.1-c1-0-13
Degree $2$
Conductor $1856$
Sign $1$
Analytic cond. $14.8202$
Root an. cond. $3.84970$
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  + 3-s − 5-s − 2·7-s − 2·9-s + 3·11-s + 13-s − 15-s + 8·17-s − 2·21-s + 4·23-s − 4·25-s − 5·27-s + 29-s − 3·31-s + 3·33-s + 2·35-s − 8·37-s + 39-s + 2·41-s + 11·43-s + 2·45-s + 13·47-s − 3·49-s + 8·51-s + 11·53-s − 3·55-s + 8·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.755·7-s − 2/3·9-s + 0.904·11-s + 0.277·13-s − 0.258·15-s + 1.94·17-s − 0.436·21-s + 0.834·23-s − 4/5·25-s − 0.962·27-s + 0.185·29-s − 0.538·31-s + 0.522·33-s + 0.338·35-s − 1.31·37-s + 0.160·39-s + 0.312·41-s + 1.67·43-s + 0.298·45-s + 1.89·47-s − 3/7·49-s + 1.12·51-s + 1.51·53-s − 0.404·55-s + 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $1$
Analytic conductor: \(14.8202\)
Root analytic conductor: \(3.84970\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1856} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.826470090\)
\(L(\frac12)\) \(\approx\) \(1.826470090\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−9.187696200797261240168787461144, −8.538714587534864706883628474190, −7.68992355213504990402216682341, −7.02533007960674119090827223190, −5.97874250935484417499511635389, −5.35705266569945906014064710560, −3.80259981482385994331528030455, −3.53944000086915839540994955829, −2.44287708620897533980700790337, −0.907460307275556682567938804358, 0.907460307275556682567938804358, 2.44287708620897533980700790337, 3.53944000086915839540994955829, 3.80259981482385994331528030455, 5.35705266569945906014064710560, 5.97874250935484417499511635389, 7.02533007960674119090827223190, 7.68992355213504990402216682341, 8.538714587534864706883628474190, 9.187696200797261240168787461144

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