↑

Properties

Label 2-1232-1232.1021-c0-0-1
Degree $2$
Conductor $1232$
Sign $0.875 + 0.482i$
Analytic cond. $0.614848$
Root an. cond. $0.784122$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.587 + 0.809i)7-s + (0.587 − 0.809i)8-s + (0.951 + 0.309i)9-s + (−0.309 − 0.951i)11-s + (−0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + 0.999·18-s + (−0.587 − 0.809i)22-s − 1.17i·23-s + (0.587 + 0.809i)25-s + 0.999i·28-s + (0.278 + 1.76i)29-s − i·32-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.587 + 0.809i)7-s + (0.587 − 0.809i)8-s + (0.951 + 0.309i)9-s + (−0.309 − 0.951i)11-s + (−0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + 0.999·18-s + (−0.587 − 0.809i)22-s − 1.17i·23-s + (0.587 + 0.809i)25-s + 0.999i·28-s + (0.278 + 1.76i)29-s − i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.875 + 0.482i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :0),\ 0.875 + 0.482i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.865318866\)
\(L(\frac12)\) \(\approx\) \(1.865318866\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 + 0.309i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (-0.951 - 0.309i)T^{2} \)
5 \( 1 + (-0.587 - 0.809i)T^{2} \)
13 \( 1 + (-0.587 + 0.809i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.951 - 0.309i)T^{2} \)
23 \( 1 + 1.17iT - T^{2} \)
29 \( 1 + (-0.278 - 1.76i)T + (-0.951 + 0.309i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (1.95 - 0.309i)T + (0.951 - 0.309i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (0.642 - 0.642i)T - iT^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 + 1.58i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.951 + 0.309i)T^{2} \)
61 \( 1 + (0.587 + 0.809i)T^{2} \)
67 \( 1 + (0.642 + 0.642i)T + iT^{2} \)
71 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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

−10.13099298717208022045046884265, −9.100622778709531610999326435724, −8.260868927870746963481120896596, −6.98312431453464783214564159385, −6.52378768227714903527470193663, −5.40546674427729166205077782113, −4.86712262124318436102262148960, −3.56651769700459456030085291234, −2.87605623009912368372891417060, −1.60230024872754652635835282050, 1.75611397364996184816687780645, 3.10451404133011712318006348524, 4.09001873629358783445089346108, 4.65154950449318336660727364156, 5.80703529786564639735295238317, 6.76790173811938826154264074259, 7.24626156038322532847174475142, 7.974498391269328844655176678483, 9.282446574469930032837150449049, 10.18640728123634574609155687695

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