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Properties

Label 2-3456-72.61-c1-0-29
Degree $2$
Conductor $3456$
Sign $0.967 - 0.254i$
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  + (5.72 + 3.30i)11-s + 7.89·17-s − 5.97i·19-s + (−2.5 + 4.33i)25-s + (−3.39 − 5.88i)41-s + (−3.82 − 2.20i)43-s + (3.5 + 6.06i)49-s + (−1.62 + 0.937i)59-s + (−14.1 + 8.18i)67-s + 15.6·73-s + (−2.44 − 1.41i)83-s + 18·89-s + (4.84 − 8.39i)97-s + 15.0i·107-s + (9 + 15.5i)113-s + ⋯
L(s)  = 1  + (1.72 + 0.996i)11-s + 1.91·17-s − 1.37i·19-s + (−0.5 + 0.866i)25-s + (−0.530 − 0.919i)41-s + (−0.583 − 0.336i)43-s + (0.5 + 0.866i)49-s + (−0.211 + 0.122i)59-s + (−1.73 + 0.999i)67-s + 1.83·73-s + (−0.268 − 0.155i)83-s + 1.90·89-s + (0.492 − 0.852i)97-s + 1.45i·107-s + (0.846 + 1.46i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $0.967 - 0.254i$
Analytic conductor: \(27.5962\)
Root analytic conductor: \(5.25321\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :1/2),\ 0.967 - 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.249639940\)
\(L(\frac12)\) \(\approx\) \(2.249639940\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.72 - 3.30i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 7.89T + 17T^{2} \)
19 \( 1 + 5.97iT - 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (3.39 + 5.88i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.82 + 2.20i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (1.62 - 0.937i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (14.1 - 8.18i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.44 + 1.41i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 + (-4.84 + 8.39i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.828632609447459843124347732248, −7.68808250337001407293020374068, −7.19250138201678595113245017303, −6.47527135586402844213936971749, −5.59341253270404647174268286201, −4.78868972455685043174925503303, −3.91619396269480455390675380757, −3.19676231165154336163376140398, −1.92577825459172937145136541638, −1.00868891429343289943695194291, 0.905752488763116180892454823120, 1.77628087847202250719899972898, 3.34178530013314072428781400699, 3.62401365782791852854195806805, 4.68928456229183487054704107131, 5.84617823006906217240282523571, 6.10349314652835465414541732299, 7.04172611096645694114591402338, 8.036993446249505441362400218564, 8.373016604108864814614770062389

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