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Properties

Label 2-18e2-36.31-c2-0-29
Degree $2$
Conductor $324$
Sign $0.854 - 0.519i$
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  + (1.65 + 1.12i)2-s + (1.45 + 3.72i)4-s + (1.80 − 3.12i)5-s + (5.40 − 3.12i)7-s + (−1.80 + 7.79i)8-s + (6.5 − 3.12i)10-s + (10.5 − 6.06i)11-s + (8 − 13.8i)13-s + (12.4 + 0.945i)14-s + (−11.7 + 10.8i)16-s − 14.4·17-s + 37.4i·19-s + (14.2 + 2.17i)20-s + (24.1 + 1.83i)22-s + (15 + 8.66i)23-s + ⋯
L(s)  = 1  + (0.825 + 0.564i)2-s + (0.363 + 0.931i)4-s + (0.360 − 0.624i)5-s + (0.772 − 0.446i)7-s + (−0.225 + 0.974i)8-s + (0.650 − 0.312i)10-s + (0.954 − 0.551i)11-s + (0.615 − 1.06i)13-s + (0.889 + 0.0675i)14-s + (−0.735 + 0.677i)16-s − 0.848·17-s + 1.97i·19-s + (0.712 + 0.108i)20-s + (1.09 + 0.0834i)22-s + (0.652 + 0.376i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.854 - 0.519i$
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: \((2,\ 324,\ (\ :1),\ 0.854 - 0.519i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.97165 + 0.832995i\)
\(L(\frac12)\) \(\approx\) \(2.97165 + 0.832995i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.65 - 1.12i)T \)
3 \( 1 \)
good5 \( 1 + (-1.80 + 3.12i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-5.40 + 3.12i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-10.5 + 6.06i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-8 + 13.8i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 14.4T + 289T^{2} \)
19 \( 1 - 37.4iT - 361T^{2} \)
23 \( 1 + (-15 - 8.66i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (25.2 + 43.7i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-5.40 - 3.12i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 26T + 1.36e3T^{2} \)
41 \( 1 + (3.60 - 6.24i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (10.8 - 6.24i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (3 - 1.73i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 68.5T + 2.80e3T^{2} \)
59 \( 1 + (66 + 38.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-54.0 - 31.2i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 62.3iT - 5.04e3T^{2} \)
73 \( 1 + 19T + 5.32e3T^{2} \)
79 \( 1 + (43.2 - 24.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-100.5 + 58.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 79.3T + 7.92e3T^{2} \)
97 \( 1 + (59.5 + 103. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.54456573042328334478734291307, −10.88260356308604109249757172805, −9.430358345283875385303548443314, −8.327112974094219319146669785282, −7.74254028958920831688388840358, −6.31141352011600886705035688831, −5.59314603648026225438929384515, −4.44532315316115764936499841648, −3.45196995980770222897671860214, −1.51518147768868096056671452768, 1.61639310371297592406930380257, 2.71616051575972780400563139406, 4.23228934585408209731725913604, 5.05413822200218538628551723749, 6.53063401222813232225393912394, 6.91091513569184527577375339600, 8.870599724525398338702100027390, 9.437690511329355316447645598298, 10.99393744203699362688407568837, 11.12186760756037032277247638772

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