Properties

Label 2-3200-40.29-c1-0-30
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

\(L(1)\) \(\approx\) \(0.6362760192\)
\(L(\frac12)\) \(\approx\) \(0.6362760192\)
\(L(\frac{3}{2})\) 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

−8.424388070197217604442950781230, −7.49919806320799352825259882459, −7.04716087456950582745593103219, −6.26052540772479174785004494437, −5.52005721384635720018472587283, −4.54821711831870523265548632201, −4.29264343856643013661026065421, −2.85842460885069053354522728851, −1.60966168125923872046140163629, −0.36406028847228352297394652487, 0.70516200040660348069073472688, 2.33440120821207507857513306277, 3.00894460282619592593678535657, 4.56623238752684872723638007069, 5.02742749764096663069953405896, 5.83704132985375205997116213277, 6.21774970982900809527807989284, 7.36404333651414282818571525407, 7.77526052302832037500498520258, 9.164682270507401831741271956473

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