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Properties

Label 2-3584-448.405-c0-0-1
Degree $2$
Conductor $3584$
Sign $0.773 + 0.634i$
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  + (−0.923 − 0.382i)7-s + (−0.923 + 0.382i)9-s + (1.08 + 0.216i)11-s + (−0.707 − 1.70i)23-s + (0.382 − 0.923i)25-s + (1.63 − 0.324i)29-s + (1.08 + 1.63i)37-s + (0.382 − 1.92i)43-s + (0.707 + 0.707i)49-s + (0.382 + 0.0761i)53-s + 63-s + (−0.0761 − 0.382i)67-s + (1.30 + 0.541i)71-s + (−0.923 − 0.617i)77-s + (−0.541 − 0.541i)79-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)7-s + (−0.923 + 0.382i)9-s + (1.08 + 0.216i)11-s + (−0.707 − 1.70i)23-s + (0.382 − 0.923i)25-s + (1.63 − 0.324i)29-s + (1.08 + 1.63i)37-s + (0.382 − 1.92i)43-s + (0.707 + 0.707i)49-s + (0.382 + 0.0761i)53-s + 63-s + (−0.0761 − 0.382i)67-s + (1.30 + 0.541i)71-s + (−0.923 − 0.617i)77-s + (−0.541 − 0.541i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.773 + 0.634i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.773 + 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.068127183\)
\(L(\frac12)\) \(\approx\) \(1.068127183\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.923 + 0.382i)T \)
good3 \( 1 + (0.923 - 0.382i)T^{2} \)
5 \( 1 + (-0.382 + 0.923i)T^{2} \)
11 \( 1 + (-1.08 - 0.216i)T + (0.923 + 0.382i)T^{2} \)
13 \( 1 + (-0.382 - 0.923i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (0.382 + 0.923i)T^{2} \)
23 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (-1.63 + 0.324i)T + (0.923 - 0.382i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1.08 - 1.63i)T + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (-0.382 + 1.92i)T + (-0.923 - 0.382i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-0.382 - 0.0761i)T + (0.923 + 0.382i)T^{2} \)
59 \( 1 + (-0.382 + 0.923i)T^{2} \)
61 \( 1 + (0.923 - 0.382i)T^{2} \)
67 \( 1 + (0.0761 + 0.382i)T + (-0.923 + 0.382i)T^{2} \)
71 \( 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
83 \( 1 + (0.382 + 0.923i)T^{2} \)
89 \( 1 + (0.707 + 0.707i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.538239580134676264426732463142, −8.154223308863013341520680952938, −6.92424916058717693296640322079, −6.48089169987875348269023210896, −5.87950526685485646140144188167, −4.67889177555949116015681814040, −4.09827975127158784762322067062, −3.03413749477570899494997355894, −2.32763638031431587320181007649, −0.73488706415226677483120727677, 1.12512123102319584129811584422, 2.53531971714350250057002713997, 3.35107889504442007055407548247, 3.97396647470101066219208292292, 5.19695788924239513066483054272, 6.00669800877600437233932265980, 6.38612894561986195117335269286, 7.29149328897651402280729041545, 8.149012848274870640456441404122, 9.048345963408762707498133751538

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