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Properties

Label 2-3584-3584.1861-c0-0-0
Degree $2$
Conductor $3584$
Sign $0.545 - 0.838i$
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  + (−0.427 + 0.903i)2-s + (−0.634 − 0.773i)4-s + (0.514 + 0.857i)7-s + (0.970 − 0.242i)8-s + (0.803 − 0.595i)9-s + (1.14 + 0.140i)11-s + (−0.995 + 0.0980i)14-s + (−0.195 + 0.980i)16-s + (0.195 + 0.980i)18-s + (−0.616 + 0.972i)22-s + (0.125 + 0.265i)23-s + (0.740 − 0.671i)25-s + (0.336 − 0.941i)28-s + (0.216 − 0.781i)29-s + (−0.803 − 0.595i)32-s + ⋯
L(s)  = 1  + (−0.427 + 0.903i)2-s + (−0.634 − 0.773i)4-s + (0.514 + 0.857i)7-s + (0.970 − 0.242i)8-s + (0.803 − 0.595i)9-s + (1.14 + 0.140i)11-s + (−0.995 + 0.0980i)14-s + (−0.195 + 0.980i)16-s + (0.195 + 0.980i)18-s + (−0.616 + 0.972i)22-s + (0.125 + 0.265i)23-s + (0.740 − 0.671i)25-s + (0.336 − 0.941i)28-s + (0.216 − 0.781i)29-s + (−0.803 − 0.595i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.545 - 0.838i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (1861, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.545 - 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.205297627\)
\(L(\frac12)\) \(\approx\) \(1.205297627\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.427 - 0.903i)T \)
7 \( 1 + (-0.514 - 0.857i)T \)
good3 \( 1 + (-0.803 + 0.595i)T^{2} \)
5 \( 1 + (-0.740 + 0.671i)T^{2} \)
11 \( 1 + (-1.14 - 0.140i)T + (0.970 + 0.242i)T^{2} \)
13 \( 1 + (-0.998 + 0.0490i)T^{2} \)
17 \( 1 + (0.195 + 0.980i)T^{2} \)
19 \( 1 + (0.336 - 0.941i)T^{2} \)
23 \( 1 + (-0.125 - 0.265i)T + (-0.634 + 0.773i)T^{2} \)
29 \( 1 + (-0.216 + 0.781i)T + (-0.857 - 0.514i)T^{2} \)
31 \( 1 + (-0.923 - 0.382i)T^{2} \)
37 \( 1 + (1.93 + 0.434i)T + (0.903 + 0.427i)T^{2} \)
41 \( 1 + (-0.0980 - 0.995i)T^{2} \)
43 \( 1 + (-0.880 + 0.443i)T + (0.595 - 0.803i)T^{2} \)
47 \( 1 + (-0.555 + 0.831i)T^{2} \)
53 \( 1 + (0.0392 + 0.141i)T + (-0.857 + 0.514i)T^{2} \)
59 \( 1 + (0.998 + 0.0490i)T^{2} \)
61 \( 1 + (0.146 - 0.989i)T^{2} \)
67 \( 1 + (-0.0661 + 0.896i)T + (-0.989 - 0.146i)T^{2} \)
71 \( 1 + (-0.0851 - 0.574i)T + (-0.956 + 0.290i)T^{2} \)
73 \( 1 + (0.471 + 0.881i)T^{2} \)
79 \( 1 + (0.0939 - 0.0284i)T + (0.831 - 0.555i)T^{2} \)
83 \( 1 + (-0.903 + 0.427i)T^{2} \)
89 \( 1 + (0.634 + 0.773i)T^{2} \)
97 \( 1 + (-0.382 + 0.923i)T^{2} \)
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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

−8.928744992235288091421852477374, −8.152482984784679906595637177472, −7.29579134912455784938371227083, −6.66218256027164321830453417075, −6.06485115019657922161610720094, −5.18456232024470568850359326908, −4.43419845790574459270583437985, −3.63793344659143169574387709495, −2.09555304475250899155408873305, −1.11952900938681849284781034270, 1.16825102778596372434004013641, 1.75899053343698748529824557928, 3.08799351586642474748571092343, 3.93099772670425993136181252381, 4.55054083703459754249775453284, 5.29235877470190922698809245628, 6.79887674436753648490490176978, 7.16633790289365537236628793390, 8.021770257746578281132591597816, 8.732474423083326438430710264781

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