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Properties

Label 2-3456-216.139-c0-0-1
Degree $2$
Conductor $3456$
Sign $-0.835 + 0.549i$
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  + (−0.984 + 0.173i)3-s + (0.939 − 0.342i)9-s + (−0.223 − 1.26i)11-s + (−0.939 + 1.62i)17-s + (−0.984 − 1.70i)19-s + (−0.939 − 0.342i)25-s + (−0.866 + 0.5i)27-s + (0.439 + 1.20i)33-s + (−0.326 + 0.118i)41-s + (0.118 + 0.673i)43-s + (0.173 − 0.984i)49-s + (0.642 − 1.76i)51-s + (1.26 + 1.50i)57-s + (−0.342 + 1.93i)59-s + (−1.20 + 0.439i)67-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)3-s + (0.939 − 0.342i)9-s + (−0.223 − 1.26i)11-s + (−0.939 + 1.62i)17-s + (−0.984 − 1.70i)19-s + (−0.939 − 0.342i)25-s + (−0.866 + 0.5i)27-s + (0.439 + 1.20i)33-s + (−0.326 + 0.118i)41-s + (0.118 + 0.673i)43-s + (0.173 − 0.984i)49-s + (0.642 − 1.76i)51-s + (1.26 + 1.50i)57-s + (−0.342 + 1.93i)59-s + (−1.20 + 0.439i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $-0.835 + 0.549i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (2623, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :0),\ -0.835 + 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2724905074\)
\(L(\frac12)\) \(\approx\) \(0.2724905074\)
\(L(1)\) 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 \( 1 + (0.984 - 0.173i)T \)
good5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (-0.173 + 0.984i)T^{2} \)
11 \( 1 + (0.223 + 1.26i)T + (-0.939 + 0.342i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.984 + 1.70i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (-0.173 - 0.984i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (-0.118 - 0.673i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.342 - 1.93i)T + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (1.20 - 0.439i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (1.62 + 0.592i)T + (0.766 + 0.642i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
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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

−8.673444378485170118089906286846, −7.68272614980441166591017201591, −6.73538001357362159418807339950, −6.15844244848556827162175288519, −5.63564539030466538923782947496, −4.54259842456877966721574338488, −4.06668115527038688421875152119, −2.88429841661822234044393913108, −1.66587044523677264107016348475, −0.17620764276159262422991927162, 1.59826181722400434346720617530, 2.41277119803446653314194383479, 3.90299461537331214472799462793, 4.57262031017533570309091169990, 5.29839552680922394285677376484, 6.08716548490742526269846984113, 6.86836884032489286637672543128, 7.42135093064978001350945223395, 8.148584125843865166317825105846, 9.259055092436486906953695385939

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