Properties

Label 2-3456-9.7-c1-0-21
Degree $2$
Conductor $3456$
Sign $0.972 + 0.231i$
Analytic cond. $27.5962$
Root an. cond. $5.25321$
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  + (−1.24 − 2.15i)5-s + (0.909 − 1.57i)7-s + (0.598 − 1.03i)11-s + (2.83 + 4.90i)13-s + 5.30·17-s + 4.55·19-s + (2.01 + 3.48i)23-s + (−0.589 + 1.02i)25-s + (−3.01 + 5.22i)29-s + (2.81 + 4.87i)31-s − 4.51·35-s − 5.18·37-s + (−4.57 − 7.92i)41-s + (−3.99 + 6.91i)43-s + (−1.39 + 2.41i)47-s + ⋯
L(s)  = 1  + (−0.555 − 0.962i)5-s + (0.343 − 0.595i)7-s + (0.180 − 0.312i)11-s + (0.785 + 1.36i)13-s + 1.28·17-s + 1.04·19-s + (0.419 + 0.727i)23-s + (−0.117 + 0.204i)25-s + (−0.559 + 0.969i)29-s + (0.505 + 0.876i)31-s − 0.763·35-s − 0.851·37-s + (−0.714 − 1.23i)41-s + (−0.608 + 1.05i)43-s + (−0.203 + 0.352i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $0.972 + 0.231i$
Analytic conductor: \(27.5962\)
Root analytic conductor: \(5.25321\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :1/2),\ 0.972 + 0.231i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.976519855\)
\(L(\frac12)\) \(\approx\) \(1.976519855\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (1.24 + 2.15i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.909 + 1.57i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.598 + 1.03i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.83 - 4.90i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.30T + 17T^{2} \)
19 \( 1 - 4.55T + 19T^{2} \)
23 \( 1 + (-2.01 - 3.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.01 - 5.22i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.81 - 4.87i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.18T + 37T^{2} \)
41 \( 1 + (4.57 + 7.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.99 - 6.91i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.39 - 2.41i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.54T + 53T^{2} \)
59 \( 1 + (1.85 + 3.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.01 - 6.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.91 - 11.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.1T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + (-4.36 + 7.56i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.89 + 15.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.455T + 89T^{2} \)
97 \( 1 + (1.01 - 1.76i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.653375449800418634795776053033, −7.81455387440005380018655167420, −7.22525014397716615148593916288, −6.38558314516405316052829236306, −5.29807357546618228594056338467, −4.83288009271390404101579219095, −3.80229583914297862443171264506, −3.34250219997410958092820624402, −1.56388421214228641295347905426, −1.00034032458578420553893193170, 0.803370108308543043722156285904, 2.19034544632605589803944484047, 3.28896654965803087428090503460, 3.56521562980352551541204232351, 4.97163646514494392587690371194, 5.56886421590038589381626202642, 6.41398291267031310798044110864, 7.19376081444172662689516594643, 8.080606657224999146753938465750, 8.205301621011246562306663662562

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