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Properties

Label 2-3584-3584.1917-c0-0-0
Degree $2$
Conductor $3584$
Sign $-0.960 + 0.278i$
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  + (0.514 − 0.857i)2-s + (−0.471 − 0.881i)4-s + (0.671 − 0.740i)7-s + (−0.998 − 0.0490i)8-s + (−0.427 + 0.903i)9-s + (−0.0487 − 1.98i)11-s + (−0.290 − 0.956i)14-s + (−0.555 + 0.831i)16-s + (0.555 + 0.831i)18-s + (−1.72 − 0.978i)22-s + (−0.968 − 1.61i)23-s + (−0.989 − 0.146i)25-s + (−0.970 − 0.242i)28-s + (0.545 + 1.41i)29-s + (0.427 + 0.903i)32-s + ⋯
L(s)  = 1  + (0.514 − 0.857i)2-s + (−0.471 − 0.881i)4-s + (0.671 − 0.740i)7-s + (−0.998 − 0.0490i)8-s + (−0.427 + 0.903i)9-s + (−0.0487 − 1.98i)11-s + (−0.290 − 0.956i)14-s + (−0.555 + 0.831i)16-s + (0.555 + 0.831i)18-s + (−1.72 − 0.978i)22-s + (−0.968 − 1.61i)23-s + (−0.989 − 0.146i)25-s + (−0.970 − 0.242i)28-s + (0.545 + 1.41i)29-s + (0.427 + 0.903i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $-0.960 + 0.278i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (1917, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ -0.960 + 0.278i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.329815282\)
\(L(\frac12)\) \(\approx\) \(1.329815282\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.514 + 0.857i)T \)
7 \( 1 + (-0.671 + 0.740i)T \)
good3 \( 1 + (0.427 - 0.903i)T^{2} \)
5 \( 1 + (0.989 + 0.146i)T^{2} \)
11 \( 1 + (0.0487 + 1.98i)T + (-0.998 + 0.0490i)T^{2} \)
13 \( 1 + (-0.803 - 0.595i)T^{2} \)
17 \( 1 + (0.555 + 0.831i)T^{2} \)
19 \( 1 + (-0.970 - 0.242i)T^{2} \)
23 \( 1 + (0.968 + 1.61i)T + (-0.471 + 0.881i)T^{2} \)
29 \( 1 + (-0.545 - 1.41i)T + (-0.740 + 0.671i)T^{2} \)
31 \( 1 + (-0.382 - 0.923i)T^{2} \)
37 \( 1 + (0.0473 + 0.0130i)T + (0.857 + 0.514i)T^{2} \)
41 \( 1 + (-0.956 + 0.290i)T^{2} \)
43 \( 1 + (-0.790 + 0.177i)T + (0.903 - 0.427i)T^{2} \)
47 \( 1 + (-0.980 - 0.195i)T^{2} \)
53 \( 1 + (-0.588 + 1.52i)T + (-0.740 - 0.671i)T^{2} \)
59 \( 1 + (0.803 - 0.595i)T^{2} \)
61 \( 1 + (0.941 - 0.336i)T^{2} \)
67 \( 1 + (0.358 + 0.252i)T + (0.336 + 0.941i)T^{2} \)
71 \( 1 + (1.19 + 0.427i)T + (0.773 + 0.634i)T^{2} \)
73 \( 1 + (0.0980 - 0.995i)T^{2} \)
79 \( 1 + (-0.920 - 0.755i)T + (0.195 + 0.980i)T^{2} \)
83 \( 1 + (-0.857 + 0.514i)T^{2} \)
89 \( 1 + (0.471 + 0.881i)T^{2} \)
97 \( 1 + (0.923 - 0.382i)T^{2} \)
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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

−8.380093291855937606836844689472, −7.992089758677959516800189380481, −6.72712395472833233586863132448, −5.89881909735513311373405351573, −5.28790928498015507538584763467, −4.46350502282048693530774033218, −3.68450013704320285360489889420, −2.83149588020293604017768513328, −1.88475788097132280129499367251, −0.63347333294774176165187815624, 1.84754545540299222548797515078, 2.79106353892891439537421777973, 4.02132199610019846039329069582, 4.45898083582921209703609830349, 5.52400975058967774630974232179, 5.92115131705762334368026489734, 6.83399422007472620918101282604, 7.68895665677948150239815425206, 7.969031305969208193092350522590, 9.149332552937760214017563483803

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