Properties

Label 2-1232-1.1-c1-0-4
Degree $2$
Conductor $1232$
Sign $1$
Analytic cond. $9.83756$
Root an. cond. $3.13649$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 7-s − 3·9-s + 11-s + 2·13-s − 4·17-s + 6·19-s − 4·23-s + 11·25-s − 2·29-s + 2·31-s − 4·35-s + 10·37-s + 4·41-s + 8·43-s + 12·45-s − 2·47-s + 49-s + 6·53-s − 4·55-s + 12·59-s − 14·61-s − 3·63-s − 8·65-s + 12·67-s + 8·71-s + 4·73-s + ⋯
L(s)  = 1  − 1.78·5-s + 0.377·7-s − 9-s + 0.301·11-s + 0.554·13-s − 0.970·17-s + 1.37·19-s − 0.834·23-s + 11/5·25-s − 0.371·29-s + 0.359·31-s − 0.676·35-s + 1.64·37-s + 0.624·41-s + 1.21·43-s + 1.78·45-s − 0.291·47-s + 1/7·49-s + 0.824·53-s − 0.539·55-s + 1.56·59-s − 1.79·61-s − 0.377·63-s − 0.992·65-s + 1.46·67-s + 0.949·71-s + 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.025800617\)
\(L(\frac12)\) \(\approx\) \(1.025800617\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−9.547191511291270231003976366558, −8.709999220022650477837071280929, −8.044641060012649095672055729373, −7.50508419532254388202035357201, −6.46937167496349882605824460904, −5.43816584265752477527041089177, −4.34120762878823751282665999727, −3.70746059636834037253003990581, −2.63625880211230133446344190029, −0.74508938317984456970237371818, 0.74508938317984456970237371818, 2.63625880211230133446344190029, 3.70746059636834037253003990581, 4.34120762878823751282665999727, 5.43816584265752477527041089177, 6.46937167496349882605824460904, 7.50508419532254388202035357201, 8.044641060012649095672055729373, 8.709999220022650477837071280929, 9.547191511291270231003976366558

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