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Properties

Label 2-5328-37.36-c1-0-60
Degree $2$
Conductor $5328$
Sign $0.600 + 0.799i$
Analytic cond. $42.5442$
Root an. cond. $6.52259$
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  − 3.19i·5-s + 3.28·7-s + 1.70·11-s + 1.45i·13-s + 4.77i·17-s + 0.209i·19-s − 8.73i·23-s − 5.23·25-s − 2.65i·29-s + 9.47i·31-s − 10.5i·35-s + (3.65 + 4.86i)37-s + 11.3·41-s − 8.67i·43-s + 12.6·47-s + ⋯
L(s)  = 1  − 1.43i·5-s + 1.24·7-s + 0.514·11-s + 0.403i·13-s + 1.15i·17-s + 0.0481i·19-s − 1.82i·23-s − 1.04·25-s − 0.492i·29-s + 1.70i·31-s − 1.77i·35-s + (0.600 + 0.799i)37-s + 1.77·41-s − 1.32i·43-s + 1.84·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5328\)    =    \(2^{4} \cdot 3^{2} \cdot 37\)
Sign: $0.600 + 0.799i$
Analytic conductor: \(42.5442\)
Root analytic conductor: \(6.52259\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5328} (2737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5328,\ (\ :1/2),\ 0.600 + 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.526893502\)
\(L(\frac12)\) \(\approx\) \(2.526893502\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37 \( 1 + (-3.65 - 4.86i)T \)
good5 \( 1 + 3.19iT - 5T^{2} \)
7 \( 1 - 3.28T + 7T^{2} \)
11 \( 1 - 1.70T + 11T^{2} \)
13 \( 1 - 1.45iT - 13T^{2} \)
17 \( 1 - 4.77iT - 17T^{2} \)
19 \( 1 - 0.209iT - 19T^{2} \)
23 \( 1 + 8.73iT - 23T^{2} \)
29 \( 1 + 2.65iT - 29T^{2} \)
31 \( 1 - 9.47iT - 31T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 + 8.67iT - 43T^{2} \)
47 \( 1 - 12.6T + 47T^{2} \)
53 \( 1 - 5.11T + 53T^{2} \)
59 \( 1 - 1.65iT - 59T^{2} \)
61 \( 1 + 1.90iT - 61T^{2} \)
67 \( 1 + 1.90T + 67T^{2} \)
71 \( 1 + 9.51T + 71T^{2} \)
73 \( 1 - 6.10T + 73T^{2} \)
79 \( 1 - 3.62iT - 79T^{2} \)
83 \( 1 + 3.20T + 83T^{2} \)
89 \( 1 + 10.1iT - 89T^{2} \)
97 \( 1 - 10.1iT - 97T^{2} \)
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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

−8.261882520405593512594963166865, −7.56298330961821458716819984566, −6.58740877315757498448108300054, −5.81204324683639921215910276634, −5.03998845750446748065329715046, −4.39158127214436000043256202213, −3.99928132557047307245309174328, −2.45829908922555648247699685248, −1.55197937892092414911542090374, −0.844263133238740799217383712472, 1.00956813569491063182601979288, 2.19404668791642410744259611436, 2.86199147559539084609100440836, 3.80740338951365422227992947655, 4.54228087169228508032858503663, 5.58221799823408339709139802024, 6.02321432613556553608254965060, 7.15112490431695629375641797070, 7.47639219959486929137164464085, 7.989401753161252086479405964202

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