Properties

Label 2-18e2-9.7-c7-0-22
Degree $2$
Conductor $324$
Sign $-0.173 + 0.984i$
Analytic cond. $101.212$
Root an. cond. $10.0604$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (251. + 435. i)5-s + (−776.5 + 1.34e3i)7-s + (3.26e3 − 5.65e3i)11-s + (399.5 + 691. i)13-s − 1.15e4·17-s − 3.55e4·19-s + (−5.30e4 − 9.18e4i)23-s + (−8.72e4 + 1.51e5i)25-s + (−7.44e4 + 1.28e5i)29-s + (−3.89e4 − 6.74e4i)31-s − 7.80e5·35-s + 2.37e5·37-s + (−1.89e5 − 3.27e5i)41-s + (1.31e4 − 2.26e4i)43-s + (1.95e5 − 3.38e5i)47-s + ⋯
L(s)  = 1  + (0.899 + 1.55i)5-s + (−0.855 + 1.48i)7-s + (0.740 − 1.28i)11-s + (0.0504 + 0.0873i)13-s − 0.570·17-s − 1.18·19-s + (−0.908 − 1.57i)23-s + (−1.11 + 1.93i)25-s + (−0.566 + 0.981i)29-s + (−0.234 − 0.406i)31-s − 3.07·35-s + 0.770·37-s + (−0.428 − 0.741i)41-s + (0.0251 − 0.0435i)43-s + (0.274 − 0.475i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(101.212\)
Root analytic conductor: \(10.0604\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :7/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.1315242734\)
\(L(\frac12)\) \(\approx\) \(0.1315242734\)
\(L(\frac{9}{2})\) 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 \( 1 + (-251. - 435. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (776.5 - 1.34e3i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-3.26e3 + 5.65e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + (-399.5 - 691. i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + 1.15e4T + 4.10e8T^{2} \)
19 \( 1 + 3.55e4T + 8.93e8T^{2} \)
23 \( 1 + (5.30e4 + 9.18e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (7.44e4 - 1.28e5i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 + (3.89e4 + 6.74e4i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 - 2.37e5T + 9.49e10T^{2} \)
41 \( 1 + (1.89e5 + 3.27e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + (-1.31e4 + 2.26e4i)T + (-1.35e11 - 2.35e11i)T^{2} \)
47 \( 1 + (-1.95e5 + 3.38e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 - 1.73e6T + 1.17e12T^{2} \)
59 \( 1 + (-9.77e4 - 1.69e5i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (2.11e5 - 3.66e5i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (9.75e5 + 1.68e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 - 1.61e6T + 9.09e12T^{2} \)
73 \( 1 + 3.47e6T + 1.10e13T^{2} \)
79 \( 1 + (2.33e6 - 4.05e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-9.57e5 + 1.65e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 - 2.61e6T + 4.42e13T^{2} \)
97 \( 1 + (-4.15e6 + 7.18e6i)T + (-4.03e13 - 6.99e13i)T^{2} \)
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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

−10.24014285099714035413538715235, −9.151786529257745978632345678593, −8.573553058953223687418303915021, −6.83014640818248770575936564899, −6.23226886864861624239414140399, −5.72569717626778036259249239178, −3.76734082338335979321394098942, −2.70594108353856874866530137974, −2.10643601620195645948478555221, −0.02745322976082743103377431043, 1.13170004712335347984809656492, 1.99522294159331683082085493065, 3.98755934208019445520064829203, 4.48584135144476828616858714241, 5.82858998345894459654138474409, 6.76566935370961925267546585360, 7.81294932695099407008732706965, 9.072280141625621502413993514434, 9.704420779664147451339606101746, 10.29539229895570780962688776126

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