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Properties

Label 2-3564-9.7-c1-0-29
Degree $2$
Conductor $3564$
Sign $0.342 + 0.939i$
Analytic cond. $28.4586$
Root an. cond. $5.33466$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.665 − 1.15i)5-s + (2.41 − 4.18i)7-s + (0.5 − 0.866i)11-s + (0.299 + 0.518i)13-s + 7.47·17-s + 2.41·19-s + (0.764 + 1.32i)23-s + (1.61 − 2.79i)25-s + (−1.43 + 2.48i)29-s + (2.98 + 5.17i)31-s − 6.43·35-s + 9.73·37-s + (3.01 + 5.22i)41-s + (1.64 − 2.84i)43-s + (−2.30 + 3.99i)47-s + ⋯
L(s)  = 1  + (−0.297 − 0.515i)5-s + (0.914 − 1.58i)7-s + (0.150 − 0.261i)11-s + (0.0829 + 0.143i)13-s + 1.81·17-s + 0.554·19-s + (0.159 + 0.276i)23-s + (0.323 − 0.559i)25-s + (−0.266 + 0.461i)29-s + (0.536 + 0.929i)31-s − 1.08·35-s + 1.60·37-s + (0.470 + 0.815i)41-s + (0.250 − 0.434i)43-s + (−0.336 + 0.583i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3564\)    =    \(2^{2} \cdot 3^{4} \cdot 11\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(28.4586\)
Root analytic conductor: \(5.33466\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3564} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3564,\ (\ :1/2),\ 0.342 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.307185535\)
\(L(\frac12)\) \(\approx\) \(2.307185535\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.665 + 1.15i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.41 + 4.18i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (-0.299 - 0.518i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 7.47T + 17T^{2} \)
19 \( 1 - 2.41T + 19T^{2} \)
23 \( 1 + (-0.764 - 1.32i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.43 - 2.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.98 - 5.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.73T + 37T^{2} \)
41 \( 1 + (-3.01 - 5.22i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.64 + 2.84i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.30 - 3.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.59T + 53T^{2} \)
59 \( 1 + (-4.74 - 8.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.73 + 8.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.25 - 2.16i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.142T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 + (-0.309 + 0.535i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.04 + 12.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.35T + 89T^{2} \)
97 \( 1 + (-1.33 + 2.31i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.171907890634601383078058392487, −7.75309006871612003660303564351, −7.17778486159791636948063533372, −6.18748831665110229662119277149, −5.20652065526002090563758110424, −4.57971132931946878168527438686, −3.84541774018086817575005221534, −3.01500432286009993683385358924, −1.33906677932171324549610163750, −0.902075627173026569153498518488, 1.17746819947827188752329460678, 2.34607734103611285276616123978, 3.02559798460637341674413915609, 4.05107423786245694436936910172, 5.12284669114834027486432481032, 5.61576332449698017488890686229, 6.33974709823268497484828651001, 7.49647816422469917854463346733, 7.88790000682683040627577776019, 8.605813642088942233737027434363

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