Properties

Label 2-3456-9.7-c1-0-14
Degree $2$
Conductor $3456$
Sign $0.927 + 0.374i$
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  + (−1.74 − 3.01i)5-s + (−1.34 + 2.33i)7-s + (−2.84 + 4.92i)11-s + (−1.76 − 3.04i)13-s − 7.65·17-s + 2.02·19-s + (0.0370 + 0.0642i)23-s + (−3.57 + 6.18i)25-s + (2.46 − 4.26i)29-s + (3.72 + 6.44i)31-s + 9.40·35-s + 5.00·37-s + (−0.482 − 0.834i)41-s + (0.255 − 0.442i)43-s + (2.83 − 4.91i)47-s + ⋯
L(s)  = 1  + (−0.779 − 1.34i)5-s + (−0.510 + 0.883i)7-s + (−0.857 + 1.48i)11-s + (−0.488 − 0.845i)13-s − 1.85·17-s + 0.463·19-s + (0.00773 + 0.0133i)23-s + (−0.714 + 1.23i)25-s + (0.456 − 0.791i)29-s + (0.668 + 1.15i)31-s + 1.59·35-s + 0.823·37-s + (−0.0752 − 0.130i)41-s + (0.0389 − 0.0674i)43-s + (0.413 − 0.716i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $0.927 + 0.374i$
Analytic conductor: \(27.5962\)
Root analytic conductor: \(5.25321\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :1/2),\ 0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9006880902\)
\(L(\frac12)\) \(\approx\) \(0.9006880902\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.74 + 3.01i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.34 - 2.33i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.84 - 4.92i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.76 + 3.04i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.65T + 17T^{2} \)
19 \( 1 - 2.02T + 19T^{2} \)
23 \( 1 + (-0.0370 - 0.0642i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.46 + 4.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.72 - 6.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.00T + 37T^{2} \)
41 \( 1 + (0.482 + 0.834i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.255 + 0.442i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.83 + 4.91i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + (4.47 + 7.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.46 + 2.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.56 + 2.71i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.19T + 71T^{2} \)
73 \( 1 - 5.21T + 73T^{2} \)
79 \( 1 + (0.716 - 1.24i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.74 + 3.01i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 + (3.50 - 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.586854498635634041291782335767, −7.88837262612039616108098275542, −7.21878769838415093051130678548, −6.28740178638338360598933172528, −5.21024401894322169145758416766, −4.83252567251934406289050067302, −4.11697171312819552989467878441, −2.81146811164743128344802797548, −2.06646964538614898770278714334, −0.50976457807393461283039191112, 0.55635342173966332077995012338, 2.46027542557850100431038961366, 3.02972142668651868034613238783, 3.93324989609313494862208582010, 4.54517495607440742385456608179, 5.86788412837524744250141183422, 6.57445689878267277017262725932, 7.12415927343660330958100022070, 7.71684266060638658149191875716, 8.558097555674328765064131759861

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