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Properties

Label 2-3456-72.11-c1-0-3
Degree $2$
Conductor $3456$
Sign $-0.885 - 0.463i$
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  + (0.158 + 0.275i)5-s + (−2.93 − 1.69i)7-s + (0.642 + 0.370i)11-s + (−2.59 + 1.5i)13-s − 4.24i·17-s + 2.57·19-s + (2.02 + 3.50i)23-s + (2.44 − 4.24i)25-s + (4.01 − 6.94i)29-s + (−4.24 + 2.45i)31-s − 1.07i·35-s + 7.34i·37-s + (0.825 − 0.476i)41-s + (−3.50 + 6.06i)43-s + (−5.15 + 8.93i)47-s + ⋯
L(s)  = 1  + (0.0710 + 0.123i)5-s + (−1.10 − 0.639i)7-s + (0.193 + 0.111i)11-s + (−0.720 + 0.416i)13-s − 1.02i·17-s + 0.589·19-s + (0.421 + 0.730i)23-s + (0.489 − 0.848i)25-s + (0.745 − 1.29i)29-s + (−0.762 + 0.440i)31-s − 0.181i·35-s + 1.20i·37-s + (0.128 − 0.0744i)41-s + (−0.533 + 0.924i)43-s + (−0.752 + 1.30i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 - 0.463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.885 - 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $-0.885 - 0.463i$
Analytic conductor: \(27.5962\)
Root analytic conductor: \(5.25321\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :1/2),\ -0.885 - 0.463i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1799226575\)
\(L(\frac12)\) \(\approx\) \(0.1799226575\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.158 - 0.275i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.93 + 1.69i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.642 - 0.370i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.59 - 1.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.24iT - 17T^{2} \)
19 \( 1 - 2.57T + 19T^{2} \)
23 \( 1 + (-2.02 - 3.50i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.01 + 6.94i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.24 - 2.45i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.34iT - 37T^{2} \)
41 \( 1 + (-0.825 + 0.476i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.50 - 6.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.15 - 8.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.51T + 53T^{2} \)
59 \( 1 + (3.50 - 2.02i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.79 + 4.5i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.36 + 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 + 6.44T + 73T^{2} \)
79 \( 1 + (8.29 + 4.78i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.21 + 1.27i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.6iT - 89T^{2} \)
97 \( 1 + (5.62 - 9.74i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.195413439294269030811515788361, −8.012324804635618407023502144713, −7.32928810594224192202482525363, −6.70180374213723463625159977563, −6.11656462104768656455410985602, −4.97080075031761910199896201432, −4.36458530263601617730359525938, −3.23533564675409758391144351280, −2.71590156486951611121979863999, −1.25045524145073852638542602688, 0.05558751881907821841637125275, 1.57086379368752640300409908761, 2.79059753722903158900285715944, 3.35977630508768964368068217290, 4.41064709331463625218173949352, 5.47659422631321957715510662102, 5.85410398176732363085801538328, 6.93274738975254753697037762186, 7.29645141325187307058815539397, 8.611755846651761682294784640692

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