Properties

Label 2-1216-1.1-c3-0-42
Degree $2$
Conductor $1216$
Sign $1$
Analytic cond. $71.7463$
Root an. cond. $8.47032$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 8·5-s − 17·7-s − 26·9-s + 70·11-s + 61·13-s + 8·15-s + 83·17-s − 19·19-s − 17·21-s + 115·23-s − 61·25-s − 53·27-s − 279·29-s − 72·31-s + 70·33-s − 136·35-s + 34·37-s + 61·39-s + 108·41-s + 192·43-s − 208·45-s − 392·47-s − 54·49-s + 83·51-s − 131·53-s + 560·55-s + ⋯
L(s)  = 1  + 0.192·3-s + 0.715·5-s − 0.917·7-s − 0.962·9-s + 1.91·11-s + 1.30·13-s + 0.137·15-s + 1.18·17-s − 0.229·19-s − 0.176·21-s + 1.04·23-s − 0.487·25-s − 0.377·27-s − 1.78·29-s − 0.417·31-s + 0.369·33-s − 0.656·35-s + 0.151·37-s + 0.250·39-s + 0.411·41-s + 0.680·43-s − 0.689·45-s − 1.21·47-s − 0.157·49-s + 0.227·51-s − 0.339·53-s + 1.37·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $1$
Analytic conductor: \(71.7463\)
Root analytic conductor: \(8.47032\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1216} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.690172977\)
\(L(\frac12)\) \(\approx\) \(2.690172977\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + p T \)
good3 \( 1 - T + p^{3} T^{2} \)
5 \( 1 - 8 T + p^{3} T^{2} \)
7 \( 1 + 17 T + p^{3} T^{2} \)
11 \( 1 - 70 T + p^{3} T^{2} \)
13 \( 1 - 61 T + p^{3} T^{2} \)
17 \( 1 - 83 T + p^{3} T^{2} \)
23 \( 1 - 5 p T + p^{3} T^{2} \)
29 \( 1 + 279 T + p^{3} T^{2} \)
31 \( 1 + 72 T + p^{3} T^{2} \)
37 \( 1 - 34 T + p^{3} T^{2} \)
41 \( 1 - 108 T + p^{3} T^{2} \)
43 \( 1 - 192 T + p^{3} T^{2} \)
47 \( 1 + 392 T + p^{3} T^{2} \)
53 \( 1 + 131 T + p^{3} T^{2} \)
59 \( 1 - 609 T + p^{3} T^{2} \)
61 \( 1 + 338 T + p^{3} T^{2} \)
67 \( 1 - 461 T + p^{3} T^{2} \)
71 \( 1 - 750 T + p^{3} T^{2} \)
73 \( 1 - 1177 T + p^{3} T^{2} \)
79 \( 1 + 22 T + p^{3} T^{2} \)
83 \( 1 - 810 T + p^{3} T^{2} \)
89 \( 1 + 476 T + p^{3} T^{2} \)
97 \( 1 - 1426 T + p^{3} T^{2} \)
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

−9.276233092838817232407378248251, −8.864140576354861380061253434329, −7.76607992946098285409730474520, −6.56817419297560162083123762057, −6.14067386863665930609530442811, −5.37888289565725550576091196400, −3.71819781474324443672830356922, −3.41039964425079254379869351903, −1.94755849058392082225080345299, −0.861851611062581382759056021152, 0.861851611062581382759056021152, 1.94755849058392082225080345299, 3.41039964425079254379869351903, 3.71819781474324443672830356922, 5.37888289565725550576091196400, 6.14067386863665930609530442811, 6.56817419297560162083123762057, 7.76607992946098285409730474520, 8.864140576354861380061253434329, 9.276233092838817232407378248251

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