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Properties

Label 2-3584-3584.2309-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 + (0.930 + 0.527i)11-s + (0.0980 + 0.995i)14-s + (0.195 − 0.980i)16-s + (−0.195 − 0.980i)18-s + (1.05 + 0.182i)22-s + (1.12 + 0.401i)23-s + (−0.998 − 0.0490i)25-s + (0.427 + 0.903i)28-s + (0.168 + 1.36i)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 + (0.930 + 0.527i)11-s + (0.0980 + 0.995i)14-s + (0.195 − 0.980i)16-s + (−0.195 − 0.980i)18-s + (1.05 + 0.182i)22-s + (1.12 + 0.401i)23-s + (−0.998 − 0.0490i)25-s + (0.427 + 0.903i)28-s + (0.168 + 1.36i)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: $\chi_{3584} (2309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.838 + 0.545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.448561721\)
\(L(\frac12)\) \(\approx\) \(2.448561721\)
\(L(1)\) 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 \( 1 + (-0.930 - 0.527i)T + (0.514 + 0.857i)T^{2} \)
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 \( 1 + (-0.168 - 1.36i)T + (-0.970 + 0.242i)T^{2} \)
31 \( 1 + (-0.923 - 0.382i)T^{2} \)
37 \( 1 + (-1.00 + 1.42i)T + (-0.336 - 0.941i)T^{2} \)
41 \( 1 + (-0.995 + 0.0980i)T^{2} \)
43 \( 1 + (1.57 - 0.115i)T + (0.989 - 0.146i)T^{2} \)
47 \( 1 + (0.555 - 0.831i)T^{2} \)
53 \( 1 + (-0.0767 + 0.622i)T + (-0.970 - 0.242i)T^{2} \)
59 \( 1 + (-0.671 - 0.740i)T^{2} \)
61 \( 1 + (-0.595 - 0.803i)T^{2} \)
67 \( 1 + (0.0461 + 0.139i)T + (-0.803 + 0.595i)T^{2} \)
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

−9.000854988741501353860779849000, −7.76845721078607246085204621153, −6.76025931585106438437780999624, −6.48449947527512806947263932963, −5.56117249860276643390768764002, −4.90812341607264798955386956280, −3.85620595461313823538369008827, −3.33913901751239952428712718764, −2.28973360729678918279134400031, −1.30507108513354877774865713948, 1.38927573957352183490344107882, 2.59335770042550458503561373508, 3.51353965810909803914039825933, 4.28412804372536624887299604403, 4.85656820608531111942848572878, 5.86026137471055848323056248269, 6.54829512615864364818736709449, 7.15821466223640134098700377792, 7.949128061242933193431874310631, 8.454907892704401589410931397160

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