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Properties

Label 2-3584-3584.2365-c0-0-0
Degree $2$
Conductor $3584$
Sign $0.482 - 0.876i$
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  + (−0.970 + 0.242i)2-s + (0.881 − 0.471i)4-s + (0.998 − 0.0490i)7-s + (−0.740 + 0.671i)8-s + (0.336 + 0.941i)9-s + (−0.191 + 0.497i)11-s + (−0.956 + 0.290i)14-s + (0.555 − 0.831i)16-s + (−0.555 − 0.831i)18-s + (0.0652 − 0.529i)22-s + (0.829 + 0.207i)23-s + (0.595 + 0.803i)25-s + (0.857 − 0.514i)28-s + (−0.432 − 0.454i)29-s + (−0.336 + 0.941i)32-s + ⋯
L(s)  = 1  + (−0.970 + 0.242i)2-s + (0.881 − 0.471i)4-s + (0.998 − 0.0490i)7-s + (−0.740 + 0.671i)8-s + (0.336 + 0.941i)9-s + (−0.191 + 0.497i)11-s + (−0.956 + 0.290i)14-s + (0.555 − 0.831i)16-s + (−0.555 − 0.831i)18-s + (0.0652 − 0.529i)22-s + (0.829 + 0.207i)23-s + (0.595 + 0.803i)25-s + (0.857 − 0.514i)28-s + (−0.432 − 0.454i)29-s + (−0.336 + 0.941i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.482 - 0.876i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (2365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.482 - 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9404221211\)
\(L(\frac12)\) \(\approx\) \(0.9404221211\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.970 - 0.242i)T \)
7 \( 1 + (-0.998 + 0.0490i)T \)
good3 \( 1 + (-0.336 - 0.941i)T^{2} \)
5 \( 1 + (-0.595 - 0.803i)T^{2} \)
11 \( 1 + (0.191 - 0.497i)T + (-0.740 - 0.671i)T^{2} \)
13 \( 1 + (0.989 - 0.146i)T^{2} \)
17 \( 1 + (-0.555 - 0.831i)T^{2} \)
19 \( 1 + (0.857 - 0.514i)T^{2} \)
23 \( 1 + (-0.829 - 0.207i)T + (0.881 + 0.471i)T^{2} \)
29 \( 1 + (0.432 + 0.454i)T + (-0.0490 + 0.998i)T^{2} \)
31 \( 1 + (-0.382 - 0.923i)T^{2} \)
37 \( 1 + (0.442 - 0.567i)T + (-0.242 - 0.970i)T^{2} \)
41 \( 1 + (0.290 + 0.956i)T^{2} \)
43 \( 1 + (0.0483 + 0.00839i)T + (0.941 + 0.336i)T^{2} \)
47 \( 1 + (0.980 + 0.195i)T^{2} \)
53 \( 1 + (1.34 - 1.41i)T + (-0.0490 - 0.998i)T^{2} \)
59 \( 1 + (-0.989 - 0.146i)T^{2} \)
61 \( 1 + (0.427 - 0.903i)T^{2} \)
67 \( 1 + (-0.333 - 0.0749i)T + (0.903 + 0.427i)T^{2} \)
71 \( 1 + (0.661 + 1.39i)T + (-0.634 + 0.773i)T^{2} \)
73 \( 1 + (-0.995 - 0.0980i)T^{2} \)
79 \( 1 + (0.186 - 0.226i)T + (-0.195 - 0.980i)T^{2} \)
83 \( 1 + (0.242 - 0.970i)T^{2} \)
89 \( 1 + (-0.881 + 0.471i)T^{2} \)
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

−8.783160771191133862595734342610, −8.068450973173001938818820568586, −7.47317303287273631573411173724, −7.03003578861360189852365405153, −5.92465943868294225674016991218, −5.10064233692157465361359406469, −4.56014979764738198757400822647, −3.09839477205407708510299221192, −2.05556891824598374084679988189, −1.34702740780134304653531034456, 0.833016859437905675037022794681, 1.83576307381331347062819786837, 2.91767765899775388750303068457, 3.75667950803676755009900739876, 4.78258658631382519382210605795, 5.76040606410124050083717865845, 6.63718717130959893485318105954, 7.22335380201724984538005441410, 8.059082486023193347930518276387, 8.673584425189959291517274113209

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