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Properties

Label 2-3456-9.4-c1-0-4
Degree $2$
Conductor $3456$
Sign $-0.589 - 0.807i$
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  + (0.705 − 1.22i)5-s + (1.17 + 2.02i)7-s + (1.30 + 2.25i)11-s + (−1.26 + 2.18i)13-s − 4.94·17-s − 1.00·19-s + (1.50 − 2.60i)23-s + (1.50 + 2.60i)25-s + (−0.0708 − 0.122i)29-s + (−4.77 + 8.26i)31-s + 3.30·35-s − 9.00·37-s + (−4.33 + 7.50i)41-s + (−3.15 − 5.46i)43-s + (3.24 + 5.62i)47-s + ⋯
L(s)  = 1  + (0.315 − 0.546i)5-s + (0.442 + 0.766i)7-s + (0.392 + 0.679i)11-s + (−0.350 + 0.606i)13-s − 1.19·17-s − 0.231·19-s + (0.314 − 0.544i)23-s + (0.300 + 0.521i)25-s + (−0.0131 − 0.0228i)29-s + (−0.856 + 1.48i)31-s + 0.558·35-s − 1.48·37-s + (−0.676 + 1.17i)41-s + (−0.481 − 0.833i)43-s + (0.473 + 0.820i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.589 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $-0.589 - 0.807i$
Analytic conductor: \(27.5962\)
Root analytic conductor: \(5.25321\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :1/2),\ -0.589 - 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.114983194\)
\(L(\frac12)\) \(\approx\) \(1.114983194\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.705 + 1.22i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.17 - 2.02i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.30 - 2.25i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.26 - 2.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.94T + 17T^{2} \)
19 \( 1 + 1.00T + 19T^{2} \)
23 \( 1 + (-1.50 + 2.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0708 + 0.122i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.77 - 8.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.00T + 37T^{2} \)
41 \( 1 + (4.33 - 7.50i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.15 + 5.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.24 - 5.62i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.02T + 53T^{2} \)
59 \( 1 + (-5.64 + 9.77i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.45 - 5.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.154 + 0.268i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.24T + 71T^{2} \)
73 \( 1 + 6.78T + 73T^{2} \)
79 \( 1 + (4.99 + 8.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.47 + 6.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + (-7.44 - 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.795249494508015514397064165065, −8.456926291095101421980355342468, −7.12583746078956659095000537577, −6.80045918735081735565689798171, −5.75122446272162807262054220123, −4.90500808897295585555967810656, −4.54430972879685125332758757709, −3.31208712679296805744067848035, −2.12617938448890692177753571735, −1.55765648753198576954938538796, 0.31032443936028423445467270601, 1.67822675366312809461411220566, 2.67556274932062418005694160058, 3.65234293570737440750333989964, 4.39194997320227720885829605431, 5.36307607184818348988928414218, 6.10773922828574981133309667806, 6.98355785912614983632600909720, 7.39579143398865551733783807643, 8.437584676903845015418531192461

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