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Properties

Label 2-3456-216.5-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  + (0.939 + 0.342i)3-s + (0.766 + 0.642i)9-s + (−0.326 − 0.118i)11-s + (1.11 + 0.642i)17-s + (0.592 − 0.342i)19-s + (−0.766 + 0.642i)25-s + (0.500 + 0.866i)27-s + (−0.266 − 0.223i)33-s + (−0.439 + 0.524i)41-s + (0.439 − 1.20i)43-s + (0.939 − 0.342i)49-s + (0.826 + 0.984i)51-s + (0.673 − 0.118i)57-s + (−1.76 + 0.642i)59-s + (1.26 − 1.50i)67-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)3-s + (0.766 + 0.642i)9-s + (−0.326 − 0.118i)11-s + (1.11 + 0.642i)17-s + (0.592 − 0.342i)19-s + (−0.766 + 0.642i)25-s + (0.500 + 0.866i)27-s + (−0.266 − 0.223i)33-s + (−0.439 + 0.524i)41-s + (0.439 − 1.20i)43-s + (0.939 − 0.342i)49-s + (0.826 + 0.984i)51-s + (0.673 − 0.118i)57-s + (−1.76 + 0.642i)59-s + (1.26 − 1.50i)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: $\chi_{3456} (3137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :0),\ 0.835 - 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.839822370\)
\(L(\frac12)\) \(\approx\) \(1.839822370\)
\(L(1)\) 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 \( 1 + (-0.939 - 0.342i)T \)
good5 \( 1 + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (-0.939 + 0.342i)T^{2} \)
11 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.592 + 0.342i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.939 + 0.342i)T^{2} \)
29 \( 1 + (0.173 - 0.984i)T^{2} \)
31 \( 1 + (-0.939 - 0.342i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.439 - 0.524i)T + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.439 + 1.20i)T + (-0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
89 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
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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

−8.898580643539482986344047166499, −7.986315266095984274318202105557, −7.65786819527176322051928374760, −6.76452379008335175510463972955, −5.67176576253548723229255448466, −5.05166491900220258037645000284, −3.97433675231108028291227363332, −3.38645751349150883605678136732, −2.47979527686125825064005817604, −1.41552265504584846256134814465, 1.13726991128587964685262064377, 2.26897915513358424499699971765, 3.09798173644066627293127052048, 3.85714298737583302493773411954, 4.83141978404670656736060841012, 5.74870643746326009638834908102, 6.57856570790907979321653634374, 7.56824521094165115957195411984, 7.77558952786415141855910235405, 8.630127342478383599386024483237

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