Properties

Label 2-3200-200.37-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 + (0.896 + 0.142i)13-s + (−1.76 + 0.896i)17-s + (0.809 + 0.587i)25-s + (0.5 + 1.53i)29-s + (0.309 − 1.95i)37-s + (−1.53 + 1.11i)41-s + (0.309 + 0.951i)45-s i·49-s + (0.809 − 1.58i)53-s + (−0.363 + 0.5i)61-s + (0.809 + 0.412i)65-s + (−0.896 + 0.142i)73-s + (−0.309 + 0.951i)81-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)5-s + (0.587 + 0.809i)9-s + (0.896 + 0.142i)13-s + (−1.76 + 0.896i)17-s + (0.809 + 0.587i)25-s + (0.5 + 1.53i)29-s + (0.309 − 1.95i)37-s + (−1.53 + 1.11i)41-s + (0.309 + 0.951i)45-s i·49-s + (0.809 − 1.58i)53-s + (−0.363 + 0.5i)61-s + (0.809 + 0.412i)65-s + (−0.896 + 0.142i)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: $\chi_{3200} (3137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ 0.684 - 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.542433431\)
\(L(\frac12)\) \(\approx\) \(1.542433431\)
\(L(1)\) 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 \( 1 + (-0.896 - 0.142i)T + (0.951 + 0.309i)T^{2} \)
17 \( 1 + (1.76 - 0.896i)T + (0.587 - 0.809i)T^{2} \)
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 \( 1 + (-0.309 + 1.95i)T + (-0.951 - 0.309i)T^{2} \)
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 \( 1 + (-0.809 + 1.58i)T + (-0.587 - 0.809i)T^{2} \)
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 \( 1 + (0.896 - 0.142i)T + (0.951 - 0.309i)T^{2} \)
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 \( 1 + (0.142 - 0.278i)T + (-0.587 - 0.809i)T^{2} \)
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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

−8.727146995308200472057641120842, −8.520961612356084321238471350587, −7.23112017668530620485462132281, −6.72075262256837512559005734315, −5.99592970667031329937516755013, −5.14552965277944324194331920234, −4.35772636762039641551769448510, −3.38283107348994227472426037544, −2.18147198244407397291609807311, −1.60842696854187760692360529991, 1.00176787655974913117146060524, 2.10219786304855690986400064380, 3.06768138586088866772665989695, 4.23757910156487224304790595559, 4.79813026587707380752505612964, 5.90666254266253228148474384820, 6.45162545666089834369227336558, 7.02921019544704755726196527546, 8.175052354192274336983341805327, 8.908909039145596992389405141287

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