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Properties

Label 2-3584-224.125-c0-0-2
Degree $2$
Conductor $3584$
Sign $0.831 - 0.555i$
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 + (1.70 − 0.707i)11-s + (−1 − i)23-s + (−0.707 + 0.707i)25-s + (1.70 + 0.707i)29-s + (0.292 + 0.707i)37-s + (−0.707 + 0.292i)43-s − 1.00i·49-s + (0.707 − 0.292i)53-s − 1.00·63-s + (0.707 + 0.292i)67-s + (−0.707 + 1.70i)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 + (1.70 − 0.707i)11-s + (−1 − i)23-s + (−0.707 + 0.707i)25-s + (1.70 + 0.707i)29-s + (0.292 + 0.707i)37-s + (−0.707 + 0.292i)43-s − 1.00i·49-s + (0.707 − 0.292i)53-s − 1.00·63-s + (0.707 + 0.292i)67-s + (−0.707 + 1.70i)77-s + 1.41i·79-s + 1.00i·81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.831 - 0.555i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.831 - 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.358926394\)
\(L(\frac12)\) \(\approx\) \(1.358926394\)
\(L(1)\) 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 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
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 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.707 - 0.707i)T^{2} \)
67 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
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

−8.607065375627503786540291949647, −8.373174555377954506761407160403, −7.16441750588894689011853466646, −6.50975507815451510557413332421, −6.01079508288955317946000409826, −4.99501887569020538239016031264, −4.12800688964829162734452772302, −3.37137158014685995945886600951, −2.33279421103037772578154333066, −1.25530732609573155675326774512, 0.953420739465735330293491649141, 2.00534637124756331419519838164, 3.41951711683495666933810492672, 4.03498371271176747974011485737, 4.52660610186374680146439142074, 5.98178813957395323848741532422, 6.48618743494455416457366728523, 7.06669211144725748826969305326, 7.79164654355167949247171506669, 8.796828482429830608201702639802

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