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Properties

Label 2-3200-200.13-c0-0-1
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

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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

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

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: $\chi_{3200} (2113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ 0.728 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.411354577\)
\(L(\frac12)\) \(\approx\) \(1.411354577\)
\(L(1)\) 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

−8.836835524597299596222750400987, −8.017020126232995486257334859814, −7.31314743931621969018464562745, −6.14629207564981743680157257263, −5.94671991008212414041771138822, −5.17428467319671046747660080506, −3.90517311460152059876187266516, −3.07811874828624651049161962562, −2.31117296637407786983136546815, −0.896026578594829711902654803981, 1.51687531081595597247784871128, 2.23382739782826012951355945531, 3.29300853141968039400977298610, 4.51853735510722989318896202187, 5.06645575810980661688153033036, 5.94555987334737690863189070481, 6.58691306830030616668050220732, 7.41274747817578446035947141491, 8.320416777627944740739642384098, 9.138983840347317819625382775696

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