Properties

Label 2-3200-200.133-c0-0-1
Degree $2$
Conductor $3200$
Sign $0.844 + 0.535i$
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  + (0.587 − 0.809i)5-s + (0.951 + 0.309i)9-s + (−0.142 − 0.278i)13-s + (0.896 − 0.142i)17-s + (−0.309 − 0.951i)25-s + (−0.5 + 0.363i)29-s + (0.809 − 0.412i)37-s + (−0.363 + 1.11i)41-s + (0.809 − 0.587i)45-s + i·49-s + (0.309 − 1.95i)53-s + (−1.53 + 0.5i)61-s + (−0.309 − 0.0489i)65-s + (−0.142 + 0.278i)73-s + (0.809 + 0.587i)81-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)5-s + (0.951 + 0.309i)9-s + (−0.142 − 0.278i)13-s + (0.896 − 0.142i)17-s + (−0.309 − 0.951i)25-s + (−0.5 + 0.363i)29-s + (0.809 − 0.412i)37-s + (−0.363 + 1.11i)41-s + (0.809 − 0.587i)45-s + i·49-s + (0.309 − 1.95i)53-s + (−1.53 + 0.5i)61-s + (−0.309 − 0.0489i)65-s + (−0.142 + 0.278i)73-s + (0.809 + 0.587i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $0.844 + 0.535i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (833, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ 0.844 + 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.535554039\)
\(L(\frac12)\) \(\approx\) \(1.535554039\)
\(L(1)\) 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 \( 1 + (-0.587 + 0.809i)T \)
good3 \( 1 + (-0.951 - 0.309i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.142 + 0.278i)T + (-0.587 + 0.809i)T^{2} \)
17 \( 1 + (-0.896 + 0.142i)T + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.587 - 0.809i)T^{2} \)
29 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 + 0.412i)T + (0.587 - 0.809i)T^{2} \)
41 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.951 - 0.309i)T^{2} \)
53 \( 1 + (-0.309 + 1.95i)T + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.951 + 0.309i)T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.142 - 0.278i)T + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.951 - 0.309i)T^{2} \)
89 \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.278 + 1.76i)T + (-0.951 - 0.309i)T^{2} \)
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

−8.842689717649587886316908821118, −7.965089829694433965207288900676, −7.44520579311409667292222986724, −6.45902644477647715307552933550, −5.66775906828682165187345455829, −4.94592735886950732673374148132, −4.27647695263755500911246797306, −3.19064604361024945934384682474, −2.01208108204101234619926427325, −1.11636081852424014317510412329, 1.37135749251995518460145877372, 2.35582403769640637779959003439, 3.37625404651925308108995833512, 4.13953179564963238509685793674, 5.17696595759226128560425953287, 6.00926090773528396167299108451, 6.66599129740851943085304256155, 7.38653901625875500025771548640, 7.967550317849732960162026461080, 9.236765117056469121325257265614

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