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Properties

Label 2-18e2-9.7-c5-0-9
Degree $2$
Conductor $324$
Sign $0.766 + 0.642i$
Analytic cond. $51.9643$
Root an. cond. $7.20863$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−118 + 204. i)7-s + (−601 − 1.04e3i)13-s − 1.43e3·19-s + (1.56e3 − 2.70e3i)25-s + (5.16e3 + 8.94e3i)31-s + 1.65e4·37-s + (1.67e3 − 2.90e3i)43-s + (−1.94e4 − 3.36e4i)49-s + (1.93e4 − 3.34e4i)61-s + (1.77e4 + 3.07e4i)67-s − 1.45e3·73-s + (5.02e4 − 8.70e4i)79-s + 2.83e5·91-s + (6.71e4 − 1.16e5i)97-s + (−7.04e4 − 1.22e5i)103-s + ⋯
L(s)  = 1  + (−0.910 + 1.57i)7-s + (−0.986 − 1.70i)13-s − 0.910·19-s + (0.5 − 0.866i)25-s + (0.964 + 1.67i)31-s + 1.98·37-s + (0.138 − 0.239i)43-s + (−1.15 − 2.00i)49-s + (0.664 − 1.15i)61-s + (0.483 + 0.837i)67-s − 0.0318·73-s + (0.906 − 1.57i)79-s + 3.59·91-s + (0.725 − 1.25i)97-s + (−0.654 − 1.13i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(51.9643\)
Root analytic conductor: \(7.20863\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :5/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.235676747\)
\(L(\frac12)\) \(\approx\) \(1.235676747\)
\(L(\frac{7}{2})\) 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 \( 1 + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (118 - 204. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (601 + 1.04e3i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + 1.41e6T^{2} \)
19 \( 1 + 1.43e3T + 2.47e6T^{2} \)
23 \( 1 + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-5.16e3 - 8.94e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 1.65e4T + 6.93e7T^{2} \)
41 \( 1 + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (-1.67e3 + 2.90e3i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + 4.18e8T^{2} \)
59 \( 1 + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.93e4 + 3.34e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.77e4 - 3.07e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 1.80e9T^{2} \)
73 \( 1 + 1.45e3T + 2.07e9T^{2} \)
79 \( 1 + (-5.02e4 + 8.70e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + 5.58e9T^{2} \)
97 \( 1 + (-6.71e4 + 1.16e5i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−10.48874855660387792895866526163, −9.791992106742207202275830264154, −8.771123095272746360928559416210, −8.005400455525024020518596532205, −6.61674221111862359330870720080, −5.78603993763266492056751332092, −4.82513099842539090008239884775, −3.08460402345986527145801217700, −2.42854857665479453080415965391, −0.42824042538507544314208966023, 0.837831091847721438811273150360, 2.41997702603844572458464687998, 3.92307873330906800661831968785, 4.55504055508093379641205298998, 6.30075341311390343684856286239, 6.96888846258784016303826747034, 7.83435164372245171197097435108, 9.328933667304452982086706881025, 9.823550753915984954209018973589, 10.85885187208129816786565523270

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