Properties

Label 2-18e2-36.31-c2-0-5
Degree $2$
Conductor $324$
Sign $-0.766 + 0.642i$
Analytic cond. $8.82836$
Root an. cond. $2.97125$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (−3.5 + 6.06i)5-s + (7.5 − 4.33i)7-s + 7.99·8-s + (−7 − 12.1i)10-s + (−7.5 + 4.33i)11-s + (−10 + 17.3i)13-s + 17.3i·14-s + (−8 + 13.8i)16-s − 8·17-s − 10.3i·19-s + 28·20-s − 17.3i·22-s + (−3 − 1.73i)23-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.700 + 1.21i)5-s + (1.07 − 0.618i)7-s + 0.999·8-s + (−0.700 − 1.21i)10-s + (−0.681 + 0.393i)11-s + (−0.769 + 1.33i)13-s + 1.23i·14-s + (−0.5 + 0.866i)16-s − 0.470·17-s − 0.546i·19-s + 1.40·20-s − 0.787i·22-s + (−0.130 − 0.0753i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ -0.766 + 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.125506 - 0.344826i\)
\(L(\frac12)\) \(\approx\) \(0.125506 - 0.344826i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
3 \( 1 \)
good5 \( 1 + (3.5 - 6.06i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-7.5 + 4.33i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (7.5 - 4.33i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (10 - 17.3i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 8T + 289T^{2} \)
19 \( 1 + 10.3iT - 361T^{2} \)
23 \( 1 + (3 + 1.73i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (5 + 8.66i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (46.5 + 26.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 10T + 1.36e3T^{2} \)
41 \( 1 + (-25 + 43.3i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (15 - 8.66i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (75 - 43.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 47T + 2.80e3T^{2} \)
59 \( 1 + (30 + 17.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-32 - 55.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-75 - 43.3i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 55T + 5.32e3T^{2} \)
79 \( 1 + (6 - 3.46i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (25.5 - 14.7i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 10T + 7.92e3T^{2} \)
97 \( 1 + (-12.5 - 21.6i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.44752554642022450264470051480, −11.05191658672016625659453079852, −10.09874068106323329291497701339, −9.050573349438660352385943567428, −7.74530911999864929148722625643, −7.38137257526998623468403854405, −6.53617644359018449271900640239, −4.98405514151574655572178477055, −4.10063735077016094832242570920, −2.08814511690127314030002257411, 0.20069326056941450593496878120, 1.75707670620755864815347834653, 3.29011477193719489409813841697, 4.75993634009444587243692881853, 5.31336368989379202832091408637, 7.60225401199629046713308736193, 8.220517853170831652394505230525, 8.771706930941856070878398950574, 9.939553417988367157964915881886, 10.98951851292882810196410845382

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