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Properties

Label 2-3584-16.13-c1-0-60
Degree $2$
Conductor $3584$
Sign $-0.923 + 0.382i$
Analytic cond. $28.6183$
Root an. cond. $5.34961$
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  + (−1.73 − 1.73i)3-s + (0.732 − 0.732i)5-s + i·7-s + 2.99i·9-s + (−2 + 2i)11-s + (1.26 + 1.26i)13-s − 2.53·15-s + 3.46·17-s + (−1.73 − 1.73i)19-s + (1.73 − 1.73i)21-s + 2.53i·23-s + 3.92i·25-s + (−4.46 − 4.46i)29-s + 0.535·31-s + 6.92·33-s + ⋯
L(s)  = 1  + (−0.999 − 0.999i)3-s + (0.327 − 0.327i)5-s + 0.377i·7-s + 0.999i·9-s + (−0.603 + 0.603i)11-s + (0.351 + 0.351i)13-s − 0.654·15-s + 0.840·17-s + (−0.397 − 0.397i)19-s + (0.377 − 0.377i)21-s + 0.528i·23-s + 0.785i·25-s + (−0.828 − 0.828i)29-s + 0.0962·31-s + 1.20·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $-0.923 + 0.382i$
Analytic conductor: \(28.6183\)
Root analytic conductor: \(5.34961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (2689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :1/2),\ -0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6819472555\)
\(L(\frac12)\) \(\approx\) \(0.6819472555\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - iT \)
good3 \( 1 + (1.73 + 1.73i)T + 3iT^{2} \)
5 \( 1 + (-0.732 + 0.732i)T - 5iT^{2} \)
11 \( 1 + (2 - 2i)T - 11iT^{2} \)
13 \( 1 + (-1.26 - 1.26i)T + 13iT^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + (1.73 + 1.73i)T + 19iT^{2} \)
23 \( 1 - 2.53iT - 23T^{2} \)
29 \( 1 + (4.46 + 4.46i)T + 29iT^{2} \)
31 \( 1 - 0.535T + 31T^{2} \)
37 \( 1 + (-6.46 + 6.46i)T - 37iT^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + (-0.535 + 0.535i)T - 43iT^{2} \)
47 \( 1 + 4.53T + 47T^{2} \)
53 \( 1 + (1 - i)T - 53iT^{2} \)
59 \( 1 + (-5.73 + 5.73i)T - 59iT^{2} \)
61 \( 1 + (8.73 + 8.73i)T + 61iT^{2} \)
67 \( 1 + (10.9 + 10.9i)T + 67iT^{2} \)
71 \( 1 + 4iT - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + (-5.73 - 5.73i)T + 83iT^{2} \)
89 \( 1 - 4.92iT - 89T^{2} \)
97 \( 1 - 18.3T + 97T^{2} \)
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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

−7.85323757996187008227361631998, −7.57047148609778278024849420812, −6.63846323035235414122061841079, −5.95549649947590858566868983990, −5.43428149646014496093818962382, −4.68655785527161616830673370809, −3.49455725739208607837940729556, −2.19508596008016339023068591843, −1.50720246992898152061941375515, −0.26214989295211281326720607727, 1.09294307360277176128461086078, 2.65965667478379691386746626182, 3.54250108737742984621511142105, 4.40585270398491580002322100527, 5.12078605561574701908688562161, 5.94112721114747809166679712325, 6.24986641319249403208293131094, 7.40171187266940156008762831381, 8.157482923499490724841031928828, 8.947950858634561500379073288408

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