Properties

Label 2-3200-40.29-c1-0-21
Degree $2$
Conductor $3200$
Sign $-0.447 - 0.894i$
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  − 2.23·3-s + 2.82i·7-s + 2.00·9-s + 2.23i·11-s + 6.32·13-s + 5i·17-s + 2.23i·19-s − 6.32i·21-s + 5.65i·23-s + 2.23·27-s − 6.32i·29-s − 5.00i·33-s − 6.32·37-s − 14.1·39-s + 3·41-s + ⋯
L(s)  = 1  − 1.29·3-s + 1.06i·7-s + 0.666·9-s + 0.674i·11-s + 1.75·13-s + 1.21i·17-s + 0.512i·19-s − 1.38i·21-s + 1.17i·23-s + 0.430·27-s − 1.17i·29-s − 0.870i·33-s − 1.03·37-s − 2.26·39-s + 0.468·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.063321241\)
\(L(\frac12)\) \(\approx\) \(1.063321241\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23T + 3T^{2} \)
7 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 - 2.23iT - 11T^{2} \)
13 \( 1 - 6.32T + 13T^{2} \)
17 \( 1 - 5iT - 17T^{2} \)
19 \( 1 - 2.23iT - 19T^{2} \)
23 \( 1 - 5.65iT - 23T^{2} \)
29 \( 1 + 6.32iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 6.32T + 37T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 8.94T + 43T^{2} \)
47 \( 1 - 2.82iT - 47T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 + 8.94iT - 59T^{2} \)
61 \( 1 + 6.32iT - 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 - 15iT - 73T^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 + T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−8.796727089730488417427776423891, −8.294655182569380450655931570440, −7.29124408117734983586732851003, −6.29692461269038374940353082708, −5.85520870842673434725548191857, −5.45367702356570102237259441948, −4.30613040222551626622500675900, −3.55773992333957888777873800859, −2.16422400914050609095237668200, −1.17977551937094503185373322435, 0.51024498801620632162235455627, 1.14720553700608517191676820822, 2.86648039038340014433134073054, 3.86238709123662853170896456152, 4.61345470602376317362772939472, 5.50039375474470517871390630572, 6.10226320554013380888754766562, 6.87690868167602804471791766859, 7.36874629376913792597507179360, 8.631043846461320897918989600299

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