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Properties

Label 2-3200-200.77-c0-0-1
Degree $2$
Conductor $3200$
Sign $0.684 + 0.728i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.951 + 0.309i)5-s + (−0.587 + 0.809i)9-s + (−0.278 − 1.76i)13-s + (0.142 − 0.278i)17-s + (0.809 − 0.587i)25-s + (0.5 − 1.53i)29-s + (0.309 − 0.0489i)37-s + (1.53 + 1.11i)41-s + (0.309 − 0.951i)45-s − i·49-s + (0.809 − 0.412i)53-s + (0.363 + 0.5i)61-s + (0.809 + 1.58i)65-s + (0.278 − 1.76i)73-s + (−0.309 − 0.951i)81-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)5-s + (−0.587 + 0.809i)9-s + (−0.278 − 1.76i)13-s + (0.142 − 0.278i)17-s + (0.809 − 0.587i)25-s + (0.5 − 1.53i)29-s + (0.309 − 0.0489i)37-s + (1.53 + 1.11i)41-s + (0.309 − 0.951i)45-s − i·49-s + (0.809 − 0.412i)53-s + (0.363 + 0.5i)61-s + (0.809 + 1.58i)65-s + (0.278 − 1.76i)73-s + (−0.309 − 0.951i)81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $0.684 + 0.728i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ 0.684 + 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8611454596\)
\(L(\frac12)\) \(\approx\) \(0.8611454596\)
\(L(1)\) 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 \( 1 + (0.951 - 0.309i)T \)
good3 \( 1 + (0.587 - 0.809i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.278 + 1.76i)T + (-0.951 + 0.309i)T^{2} \)
17 \( 1 + (-0.142 + 0.278i)T + (-0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.951 - 0.309i)T^{2} \)
29 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.0489i)T + (0.951 - 0.309i)T^{2} \)
41 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.587 - 0.809i)T^{2} \)
53 \( 1 + (-0.809 + 0.412i)T + (0.587 - 0.809i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.587 + 0.809i)T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.278 + 1.76i)T + (-0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.587 - 0.809i)T^{2} \)
89 \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-1.76 + 0.896i)T + (0.587 - 0.809i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.468406908616442160590598974611, −7.85823705431046114347194033415, −7.63443651581686307659377437055, −6.50258271844231974747592392518, −5.64958604735750515237201100204, −4.93702545966572107763287778313, −4.04468324140963730380499653137, −3.03438608557176455827680543950, −2.46754587853950075136474218024, −0.61201028299603333833511605368, 1.13245883968908674776834178519, 2.49769338831887000587190540143, 3.58691417295652220641385373025, 4.18276700272241489947510521298, 4.99772626019275995714293186367, 5.99975562659297493485732872761, 6.82082594240711813703756384934, 7.36424784891371032826396932828, 8.309988287284790126484151002659, 9.038298705677380986965994016795

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