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Properties

Label 2-3584-224.13-c0-0-1
Degree $2$
Conductor $3584$
Sign $0.555 + 0.831i$
Analytic cond. $1.78864$
Root an. cond. $1.33740$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (−0.292 − 0.707i)11-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (0.292 − 0.707i)29-s + (1.70 − 0.707i)37-s + (−0.707 − 1.70i)43-s − 1.00i·49-s + (−0.707 − 1.70i)53-s + 1.00·63-s + (0.707 − 1.70i)67-s + (0.707 + 0.292i)77-s + 1.41i·79-s + 1.00i·81-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (−0.292 − 0.707i)11-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (0.292 − 0.707i)29-s + (1.70 − 0.707i)37-s + (−0.707 − 1.70i)43-s − 1.00i·49-s + (−0.707 − 1.70i)53-s + 1.00·63-s + (0.707 − 1.70i)67-s + (0.707 + 0.292i)77-s + 1.41i·79-s + 1.00i·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.555 + 0.831i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (2113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.555 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9708000650\)
\(L(\frac12)\) \(\approx\) \(0.9708000650\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.707 + 0.707i)T^{2} \)
67 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 - T^{2} \)
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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

−8.616508497985280282738783644406, −8.092202497929530893230421368081, −6.98569125567371733398520373644, −6.32365705017853690458337977366, −5.70175724244050436504559984286, −5.00603185741370927174777646266, −3.72685142504968417407521490912, −3.10650199473350869445964997173, −2.30252273172149934569813484626, −0.62417873863026576592324344489, 1.20205910977634357760390577406, 2.65257611088873913911994785542, 3.14023172761004517891725750238, 4.45390107142037661639832817145, 4.88521595745778831048057574183, 5.94241032697833497107627595968, 6.68672465904155037285660784957, 7.36554843045136771436257716824, 8.045611378025312251328684597954, 8.864505864018625686613756459805

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