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Properties

Label 2-18e2-9.5-c8-0-31
Degree $2$
Conductor $324$
Sign $-0.642 - 0.766i$
Analytic cond. $131.990$
Root an. cond. $11.4887$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.01e3 − 3.49e3i)7-s + (1.79e4 − 3.10e4i)13-s − 2.58e5·19-s + (−1.95e5 − 3.38e5i)25-s + (9.04e5 − 1.56e6i)31-s + 5.03e5·37-s + (−1.74e6 − 3.02e6i)43-s + (−5.25e6 + 9.10e6i)49-s + (1.19e7 + 2.06e7i)61-s + (2.71e6 − 4.69e6i)67-s + 1.61e7·73-s + (9.44e6 + 1.63e7i)79-s − 1.44e8·91-s + (−8.84e7 − 1.53e8i)97-s + (−2.22e7 + 3.84e7i)103-s + ⋯
L(s)  = 1  + (−0.840 − 1.45i)7-s + (0.626 − 1.08i)13-s − 1.98·19-s + (−0.5 − 0.866i)25-s + (0.979 − 1.69i)31-s + 0.268·37-s + (−0.510 − 0.884i)43-s + (−0.911 + 1.57i)49-s + (0.860 + 1.49i)61-s + (0.134 − 0.232i)67-s + 0.569·73-s + (0.242 + 0.419i)79-s − 2.10·91-s + (−0.999 − 1.73i)97-s + (−0.197 + 0.342i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(131.990\)
Root analytic conductor: \(11.4887\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :4),\ -0.642 - 0.766i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.4733780630\)
\(L(\frac12)\) \(\approx\) \(0.4733780630\)
\(L(5)\) 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 \( 1 + (1.95e5 + 3.38e5i)T^{2} \)
7 \( 1 + (2.01e3 + 3.49e3i)T + (-2.88e6 + 4.99e6i)T^{2} \)
11 \( 1 + (1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 + (-1.79e4 + 3.10e4i)T + (-4.07e8 - 7.06e8i)T^{2} \)
17 \( 1 - 6.97e9T^{2} \)
19 \( 1 + 2.58e5T + 1.69e10T^{2} \)
23 \( 1 + (3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 + (2.50e11 - 4.33e11i)T^{2} \)
31 \( 1 + (-9.04e5 + 1.56e6i)T + (-4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 - 5.03e5T + 3.51e12T^{2} \)
41 \( 1 + (3.99e12 + 6.91e12i)T^{2} \)
43 \( 1 + (1.74e6 + 3.02e6i)T + (-5.84e12 + 1.01e13i)T^{2} \)
47 \( 1 + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 - 6.22e13T^{2} \)
59 \( 1 + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (-1.19e7 - 2.06e7i)T + (-9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-2.71e6 + 4.69e6i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 - 6.45e14T^{2} \)
73 \( 1 - 1.61e7T + 8.06e14T^{2} \)
79 \( 1 + (-9.44e6 - 1.63e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 + (1.12e15 - 1.95e15i)T^{2} \)
89 \( 1 - 3.93e15T^{2} \)
97 \( 1 + (8.84e7 + 1.53e8i)T + (-3.91e15 + 6.78e15i)T^{2} \)
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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

−9.970639523467609981163765214133, −8.572922088094647786347454738559, −7.75469643550362731630167330637, −6.65822614359063744180628316725, −5.95636965011914742950809370120, −4.36156935697647920845673272182, −3.70005543371782148132541209702, −2.41460745194497708422375330832, −0.837357573385050783306262478861, −0.11582544503056237028526641737, 1.64287966455978582990305820425, 2.61724699338215760435855471286, 3.77105350827286726438428774383, 5.01241724588277672009945684442, 6.22715716114645624361080937818, 6.65203404376633550482421383891, 8.306940146030782519466177975796, 8.925777560684071260822647622830, 9.723228351268854373709354098227, 10.87659784576660504178985328767

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