↑

Properties

Label 2-3200-200.11-c0-0-0
Degree $2$
Conductor $3200$
Sign $0.728 - 0.684i$
Analytic cond. $1.59700$
Root an. cond. $1.26372$
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  + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)9-s + (−1.11 + 1.53i)13-s + (0.5 + 1.53i)17-s + (−0.809 + 0.587i)25-s + (1.11 + 0.363i)29-s + (−0.690 + 0.951i)37-s + (−0.5 − 0.363i)41-s + (0.309 − 0.951i)45-s + 49-s + (1.80 + 0.587i)53-s + (−1.11 − 1.53i)61-s + (1.80 + 0.587i)65-s + (0.5 − 0.363i)73-s + (0.309 + 0.951i)81-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)9-s + (−1.11 + 1.53i)13-s + (0.5 + 1.53i)17-s + (−0.809 + 0.587i)25-s + (1.11 + 0.363i)29-s + (−0.690 + 0.951i)37-s + (−0.5 − 0.363i)41-s + (0.309 − 0.951i)45-s + 49-s + (1.80 + 0.587i)53-s + (−1.11 − 1.53i)61-s + (1.80 + 0.587i)65-s + (0.5 − 0.363i)73-s + (0.309 + 0.951i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $0.728 - 0.684i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (2111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ 0.728 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.123620639\)
\(L(\frac12)\) \(\approx\) \(1.123620639\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (-0.809 - 0.587i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
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

−8.784513503517522403430009737075, −8.318057407515198064223043742581, −7.41842407368623577137545272392, −6.86016765239871538833571110911, −5.85300221237560484308325077907, −4.85161044254300700107330884179, −4.46236147769698298385916384452, −3.61755813301617864176875750554, −2.14165531233013198401759591911, −1.39951522512334665133599809721, 0.72858396334323566518485710608, 2.46331467358475224460943264113, 3.06118125411316418869301878784, 3.96773002813336913466049378123, 4.95676560386892569382700941266, 5.70331246113306750785785986948, 6.72611398492718179117050911681, 7.33352826776787293940121069855, 7.70359647721285940862557551021, 8.757741185263698080183777359591

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