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Properties

Label 2-3456-216.149-c0-0-1
Degree $2$
Conductor $3456$
Sign $0.0581 + 0.998i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
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  + (0.766 − 0.642i)3-s + (0.173 − 0.984i)9-s + (1.43 − 1.20i)11-s + (−1.70 − 0.984i)17-s + (−1.11 + 0.642i)19-s + (−0.173 − 0.984i)25-s + (−0.500 − 0.866i)27-s + (0.326 − 1.85i)33-s + (1.26 + 0.223i)41-s + (1.26 + 1.50i)43-s + (−0.766 − 0.642i)49-s + (−1.93 + 0.342i)51-s + (−0.439 + 1.20i)57-s + (1.17 + 0.984i)59-s + (−0.673 − 0.118i)67-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)3-s + (0.173 − 0.984i)9-s + (1.43 − 1.20i)11-s + (−1.70 − 0.984i)17-s + (−1.11 + 0.642i)19-s + (−0.173 − 0.984i)25-s + (−0.500 − 0.866i)27-s + (0.326 − 1.85i)33-s + (1.26 + 0.223i)41-s + (1.26 + 1.50i)43-s + (−0.766 − 0.642i)49-s + (−1.93 + 0.342i)51-s + (−0.439 + 1.20i)57-s + (1.17 + 0.984i)59-s + (−0.673 − 0.118i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0581 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0581 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $0.0581 + 0.998i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (1985, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :0),\ 0.0581 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.624100582\)
\(L(\frac12)\) \(\approx\) \(1.624100582\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.766 + 0.642i)T \)
good5 \( 1 + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.766 + 0.642i)T^{2} \)
11 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (1.70 + 0.984i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.11 - 0.642i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.939 + 0.342i)T^{2} \)
31 \( 1 + (0.766 - 0.642i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-1.26 - 0.223i)T + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (-1.26 - 1.50i)T + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.673 + 0.118i)T + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
89 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
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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

−8.622798378833532867556510355772, −8.010033645891034268566009502671, −7.02686560003628714209152916079, −6.39947876537252632614234666176, −5.97587502034473360103587184067, −4.43424084649952601688869366577, −3.94833800739039182413554894192, −2.88129802925882295396905124732, −2.09440503170901028834353834924, −0.880025511393813559651973778678, 1.79210530581222532543285776499, 2.36177599981485549271954868436, 3.73555332227096551013856568734, 4.22213194637553735998870633295, 4.80897508696131677401902305872, 6.05423547009782311110331042555, 6.82562254061589860619559698603, 7.43747290673212862716707029555, 8.456386352027627078228275826359, 9.101700826379879958308419995444

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