Properties

Label 2-3584-3584.2253-c0-0-0
Degree $2$
Conductor $3584$
Sign $0.838 - 0.545i$
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  + (−0.941 − 0.336i)2-s + (0.773 + 0.634i)4-s + (0.242 + 0.970i)7-s + (−0.514 − 0.857i)8-s + (−0.146 − 0.989i)9-s + (0.832 + 1.47i)11-s + (0.0980 − 0.995i)14-s + (0.195 + 0.980i)16-s + (−0.195 + 0.980i)18-s + (−0.288 − 1.66i)22-s + (1.12 − 0.401i)23-s + (0.998 − 0.0490i)25-s + (−0.427 + 0.903i)28-s + (−1.43 − 0.177i)29-s + (0.146 − 0.989i)32-s + ⋯
L(s)  = 1  + (−0.941 − 0.336i)2-s + (0.773 + 0.634i)4-s + (0.242 + 0.970i)7-s + (−0.514 − 0.857i)8-s + (−0.146 − 0.989i)9-s + (0.832 + 1.47i)11-s + (0.0980 − 0.995i)14-s + (0.195 + 0.980i)16-s + (−0.195 + 0.980i)18-s + (−0.288 − 1.66i)22-s + (1.12 − 0.401i)23-s + (0.998 − 0.0490i)25-s + (−0.427 + 0.903i)28-s + (−1.43 − 0.177i)29-s + (0.146 − 0.989i)32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.838 - 0.545i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (2253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.838 - 0.545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8824879972\)
\(L(\frac12)\) \(\approx\) \(0.8824879972\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.941 + 0.336i)T \)
7 \( 1 + (-0.242 - 0.970i)T \)
good3 \( 1 + (0.146 + 0.989i)T^{2} \)
5 \( 1 + (-0.998 + 0.0490i)T^{2} \)
11 \( 1 + (-0.832 - 1.47i)T + (-0.514 + 0.857i)T^{2} \)
13 \( 1 + (-0.671 - 0.740i)T^{2} \)
17 \( 1 + (-0.195 + 0.980i)T^{2} \)
19 \( 1 + (-0.427 + 0.903i)T^{2} \)
23 \( 1 + (-1.12 + 0.401i)T + (0.773 - 0.634i)T^{2} \)
29 \( 1 + (1.43 + 0.177i)T + (0.970 + 0.242i)T^{2} \)
31 \( 1 + (-0.923 + 0.382i)T^{2} \)
37 \( 1 + (0.805 - 0.567i)T + (0.336 - 0.941i)T^{2} \)
41 \( 1 + (-0.995 - 0.0980i)T^{2} \)
43 \( 1 + (0.0905 - 1.22i)T + (-0.989 - 0.146i)T^{2} \)
47 \( 1 + (0.555 + 0.831i)T^{2} \)
53 \( 1 + (-1.88 + 0.232i)T + (0.970 - 0.242i)T^{2} \)
59 \( 1 + (0.671 - 0.740i)T^{2} \)
61 \( 1 + (0.595 - 0.803i)T^{2} \)
67 \( 1 + (-1.89 - 0.625i)T + (0.803 + 0.595i)T^{2} \)
71 \( 1 + (1.14 + 1.53i)T + (-0.290 + 0.956i)T^{2} \)
73 \( 1 + (0.881 + 0.471i)T^{2} \)
79 \( 1 + (0.430 - 1.41i)T + (-0.831 - 0.555i)T^{2} \)
83 \( 1 + (-0.336 - 0.941i)T^{2} \)
89 \( 1 + (-0.773 - 0.634i)T^{2} \)
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

−8.882141892482580999282248141909, −8.375014772691245746158827771368, −7.22107940777390200413728588772, −6.85649608480020116382014006833, −6.05159043222990350337901133829, −5.00807417316847156834103978441, −4.00747946101623389214311577448, −3.07421852334940435102945761381, −2.17231463341308017250376893456, −1.22909640619088960473228380640, 0.810309108112612891760239271268, 1.81322983050918424375707787732, 3.05141000882852859597198329159, 3.94353077287458240576400544143, 5.20441363788817937591850784510, 5.66175435303377352273656873258, 6.79256899029032828748715451983, 7.20078842143238077138992282497, 7.959940053322525094157037934280, 8.782557335770448124822014508538

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