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Properties

Label 2-3456-72.11-c1-0-15
Degree $2$
Conductor $3456$
Sign $-0.141 - 0.990i$
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  + (1.07 + 1.85i)5-s + (1.13 + 0.652i)7-s + (2.36 + 1.36i)11-s + (1.16 − 0.673i)13-s + 4.27i·17-s − 2.16·19-s + (1.01 + 1.76i)23-s + (0.205 − 0.355i)25-s + (−2.72 + 4.72i)29-s + (−6.09 + 3.51i)31-s + 2.79i·35-s + 10.3i·37-s + (−0.596 + 0.344i)41-s + (−1.22 + 2.12i)43-s + (6.56 − 11.3i)47-s + ⋯
L(s)  = 1  + (0.479 + 0.829i)5-s + (0.427 + 0.246i)7-s + (0.711 + 0.411i)11-s + (0.323 − 0.186i)13-s + 1.03i·17-s − 0.495·19-s + (0.212 + 0.368i)23-s + (0.0410 − 0.0711i)25-s + (−0.506 + 0.877i)29-s + (−1.09 + 0.631i)31-s + 0.472i·35-s + 1.69i·37-s + (−0.0931 + 0.0537i)41-s + (−0.187 + 0.324i)43-s + (0.957 − 1.65i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.141 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $-0.141 - 0.990i$
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: \((2,\ 3456,\ (\ :1/2),\ -0.141 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.061235104\)
\(L(\frac12)\) \(\approx\) \(2.061235104\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.07 - 1.85i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.13 - 0.652i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.36 - 1.36i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.16 + 0.673i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.27iT - 17T^{2} \)
19 \( 1 + 2.16T + 19T^{2} \)
23 \( 1 + (-1.01 - 1.76i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.72 - 4.72i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.09 - 3.51i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.3iT - 37T^{2} \)
41 \( 1 + (0.596 - 0.344i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.22 - 2.12i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.56 + 11.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.56T + 53T^{2} \)
59 \( 1 + (-8.97 + 5.18i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.68 - 4.43i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.23 + 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 + (-0.680 - 0.393i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.82 + 1.05i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.1iT - 89T^{2} \)
97 \( 1 + (8.05 - 13.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.712011572012649926284476528162, −8.197608312077418553289647794670, −7.03278213302311611619777708778, −6.72094887675512342712953550424, −5.81507001801039175391874266785, −5.12160505197163789342015895144, −4.02536298527089148007334233482, −3.31704759043055810449026332298, −2.19243928425785457667316796379, −1.43819639580255313667694232639, 0.62850295049340561584685251307, 1.61657469525554263996685061547, 2.63075640546178903706008008476, 3.94612518666286820024307732140, 4.43062185093085282273261996532, 5.52403682493603408878567546413, 5.88960338672344782447115681539, 7.03166138589328798860959232661, 7.57129321137083038791155179687, 8.664212473185384562674826004122

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