Properties

Label 2-36784-1.1-c1-0-6
Degree $2$
Conductor $36784$
Sign $1$
Analytic cond. $293.721$
Root an. cond. $17.1383$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 5-s − 3·7-s + 9-s + 4·13-s − 2·15-s − 5·17-s − 19-s − 6·21-s − 4·25-s − 4·27-s − 2·29-s − 8·31-s + 3·35-s − 10·37-s + 8·39-s − 6·41-s − 7·43-s − 45-s + 9·47-s + 2·49-s − 10·51-s − 8·53-s − 2·57-s − 14·59-s + 5·61-s − 3·63-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s + 1.10·13-s − 0.516·15-s − 1.21·17-s − 0.229·19-s − 1.30·21-s − 4/5·25-s − 0.769·27-s − 0.371·29-s − 1.43·31-s + 0.507·35-s − 1.64·37-s + 1.28·39-s − 0.937·41-s − 1.06·43-s − 0.149·45-s + 1.31·47-s + 2/7·49-s − 1.40·51-s − 1.09·53-s − 0.264·57-s − 1.82·59-s + 0.640·61-s − 0.377·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36784\)    =    \(2^{4} \cdot 11^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(293.721\)
Root analytic conductor: \(17.1383\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 36784,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.099203898\)
\(L(\frac12)\) \(\approx\) \(1.099203898\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
19 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.02770344665840, −14.22128241872096, −13.80846585146381, −13.42965883759212, −12.89002505644395, −12.50475174653259, −11.67426639817139, −11.21167423958921, −10.62072448209691, −10.05597064649239, −9.258496498788858, −9.058343626779740, −8.552420074914587, −7.974013326064728, −7.383702101764961, −6.732035400203362, −6.274233685892784, −5.603038832516039, −4.765835628743517, −3.901356811320275, −3.474114823740293, −3.232270134388618, −2.137131932255838, −1.776731004876122, −0.3358140431287355, 0.3358140431287355, 1.776731004876122, 2.137131932255838, 3.232270134388618, 3.474114823740293, 3.901356811320275, 4.765835628743517, 5.603038832516039, 6.274233685892784, 6.732035400203362, 7.383702101764961, 7.974013326064728, 8.552420074914587, 9.058343626779740, 9.258496498788858, 10.05597064649239, 10.62072448209691, 11.21167423958921, 11.67426639817139, 12.50475174653259, 12.89002505644395, 13.42965883759212, 13.80846585146381, 14.22128241872096, 15.02770344665840

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