Properties

Label 2-18e2-36.31-c2-0-13
Degree $2$
Conductor $324$
Sign $-0.813 - 0.581i$
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  + (1.58 + 1.22i)2-s + (1.00 + 3.87i)4-s + (0.540 − 0.935i)5-s + (−5.20 + 3.00i)7-s + (−3.16 + 7.34i)8-s + (2 − 0.817i)10-s + (−15.3 + 8.86i)11-s + (−6.20 + 10.7i)13-s + (−11.9 − 1.62i)14-s + (−13.9 + 7.74i)16-s + 26.3·17-s − 5.19i·19-s + (4.16 + 1.15i)20-s + (−35.1 − 4.78i)22-s + (25.8 + 14.9i)23-s + ⋯
L(s)  = 1  + (0.790 + 0.612i)2-s + (0.250 + 0.968i)4-s + (0.108 − 0.187i)5-s + (−0.744 + 0.429i)7-s + (−0.395 + 0.918i)8-s + (0.200 − 0.0817i)10-s + (−1.39 + 0.805i)11-s + (−0.477 + 0.827i)13-s + (−0.851 − 0.116i)14-s + (−0.874 + 0.484i)16-s + 1.55·17-s − 0.273i·19-s + (0.208 + 0.0578i)20-s + (−1.59 − 0.217i)22-s + (1.12 + 0.648i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.813 - 0.581i$
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: \((2,\ 324,\ (\ :1),\ -0.813 - 0.581i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.565463 + 1.76531i\)
\(L(\frac12)\) \(\approx\) \(0.565463 + 1.76531i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.58 - 1.22i)T \)
3 \( 1 \)
good5 \( 1 + (-0.540 + 0.935i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (5.20 - 3.00i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (15.3 - 8.86i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (6.20 - 10.7i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 26.3T + 289T^{2} \)
19 \( 1 + 5.19iT - 361T^{2} \)
23 \( 1 + (-25.8 - 14.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-2.16 - 3.74i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (38.8 + 22.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 20.4T + 1.36e3T^{2} \)
41 \( 1 + (-29.6 + 51.3i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (16.5 - 9.57i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-35.5 + 20.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 70.0T + 2.80e3T^{2} \)
59 \( 1 + (25.0 + 14.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (9.20 + 15.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-82.1 - 47.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 83.8iT - 5.04e3T^{2} \)
73 \( 1 - 55.8T + 5.32e3T^{2} \)
79 \( 1 + (-35.5 + 20.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-30.7 + 17.7i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 26.3T + 7.92e3T^{2} \)
97 \( 1 + (38.1 + 66.1i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.17490269736108769838711337875, −11.00433479123703969334094838235, −9.762095541607675234508696527932, −8.922969095921644808712030922357, −7.56215852632255343392649381280, −7.05361928454092486960756672514, −5.59902748318558735893760868237, −5.07382100419183662575379118299, −3.57351000632698981319495311175, −2.40182115931470279194052767160, 0.65388552550721329184128716579, 2.75573378064814627840140636132, 3.42539530671263134984636664963, 5.04373790036654060971349926472, 5.78218961872545642953853688600, 6.97137419796292928672838630229, 8.089976697401305089030248293499, 9.544690769956151635978611625298, 10.51691640323091694657598509001, 10.71693352407618215240960927066

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