Properties

Label 2-3200-8.5-c1-0-18
Degree $2$
Conductor $3200$
Sign $-1$
Analytic cond. $25.5521$
Root an. cond. $5.05491$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.16i·3-s + 4.24·7-s − 7.00·9-s + 13.4i·21-s − 1.41·23-s − 12.6i·27-s + 8.94i·29-s − 12·41-s + 3.16i·43-s − 9.89·47-s + 10.9·49-s + 13.4i·61-s − 29.6·63-s + 15.8i·67-s − 4.47i·69-s + ⋯
L(s)  = 1  + 1.82i·3-s + 1.60·7-s − 2.33·9-s + 2.92i·21-s − 0.294·23-s − 2.43i·27-s + 1.66i·29-s − 1.87·41-s + 0.482i·43-s − 1.44·47-s + 1.57·49-s + 1.71i·61-s − 3.74·63-s + 1.93i·67-s − 0.538i·69-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.614668560\)
\(L(\frac12)\) \(\approx\) \(1.614668560\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 - 3.16iT - 3T^{2} \)
7 \( 1 - 4.24T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 1.41T + 23T^{2} \)
29 \( 1 - 8.94iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 - 3.16iT - 43T^{2} \)
47 \( 1 + 9.89T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 13.4iT - 61T^{2} \)
67 \( 1 - 15.8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 9.48iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−8.927024866643178444056521243226, −8.547031697008030689501281972201, −7.84616596504854576412816619281, −6.75220449730484085054574464100, −5.52956896075894788336641543047, −5.11023260360190335735175600501, −4.48218401815191520097636881223, −3.73833360125606421002503620570, −2.82051180885537447940440090741, −1.53881792470247520344419201980, 0.47868859532977740265935942113, 1.71702779871546634814321851182, 2.03394137902881336513794194195, 3.27172293108788720407437899697, 4.60297674271971492224693455695, 5.36549483512387828763251084359, 6.20362314204847542444611779041, 6.86313756113032677674797398866, 7.71017836851461426821301897121, 8.101917750529173277558681843698

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