Properties

Label 2-3200-40.29-c1-0-20
Degree $2$
Conductor $3200$
Sign $0.316 - 0.948i$
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  − 3·9-s + 4·13-s + 2i·17-s + 4i·29-s − 12·37-s + 10·41-s + 7·49-s − 4·53-s + 12i·61-s + 6i·73-s + 9·81-s + 10·89-s + 18i·97-s + 20i·101-s + 20i·109-s + ⋯
L(s)  = 1  − 9-s + 1.10·13-s + 0.485i·17-s + 0.742i·29-s − 1.97·37-s + 1.56·41-s + 49-s − 0.549·53-s + 1.53i·61-s + 0.702i·73-s + 81-s + 1.05·89-s + 1.82i·97-s + 1.99i·101-s + 1.91i·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $0.316 - 0.948i$
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: \((2,\ 3200,\ (\ :1/2),\ 0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.394487793\)
\(L(\frac12)\) \(\approx\) \(1.394487793\)
\(L(\frac{3}{2})\) 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 \( 1 + 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 12T + 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 12iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 18iT - 97T^{2} \)
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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

−8.789588214984616936995658760511, −8.218479137035831349038979197212, −7.34920351604880914188833212294, −6.45559539554391060175783871911, −5.80951898610700459269131207622, −5.12912111247385227923289009276, −3.98987411273445258550065834144, −3.31369348684660704816412464670, −2.29588520903813057044376099748, −1.08645371298573824041980669375, 0.48192076835468992320752258759, 1.86274877259394954740602944111, 2.94353723896191981835740042222, 3.68579418290606987770970817854, 4.67889714849769342898730033339, 5.61813737571389026957901027864, 6.12899400395768050817262831828, 7.01234917133769375689332390362, 7.87229092446838130368023295305, 8.580967856813785012333115446052

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