Properties

Label 2-3200-200.197-c0-0-1
Degree $2$
Conductor $3200$
Sign $0.535 + 0.844i$
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  + (−0.587 − 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

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

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $0.535 + 0.844i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (2497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ 0.535 + 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.213664650\)
\(L(\frac12)\) \(\approx\) \(1.213664650\)
\(L(1)\) 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 \( 1 + (-0.278 - 1.76i)T + (-0.951 + 0.309i)T^{2} \)
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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

−8.638771016668306034888168859563, −8.054293042323090836143268893710, −7.27090655326517722418202704793, −6.62979104546790337516947119068, −5.47571100069955581170277931634, −4.95664229847759665160722726278, −3.91774638700157139182497981357, −3.44815721012636015932098855729, −1.92489671041596882471779467117, −0.856932412507434451914351824898, 1.32363015267642575494438961262, 2.56391745628400610134957330604, 3.45231934125202235594437846661, 4.25221985024888829643382003140, 5.01989376734087601325233688529, 6.14144605723938295091388787355, 6.77322890030007421486666914854, 7.56027125383030344581447839838, 7.967362581036670638129825445263, 8.947880342513916514148507667185

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