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Properties

Label 2-18e2-9.7-c5-0-19
Degree $2$
Conductor $324$
Sign $-0.939 + 0.342i$
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  + (−28.8 − 49.9i)5-s + (94.4 − 163. i)7-s + (352. − 609. i)11-s + (−397. − 688. i)13-s + 180.·17-s − 661.·19-s + (1.81e3 + 3.14e3i)23-s + (−97.9 + 169. i)25-s + (4.01e3 − 6.94e3i)29-s + (1.50e3 + 2.60e3i)31-s − 1.08e4·35-s − 1.52e3·37-s + (1.73e3 + 2.99e3i)41-s + (−5.90e3 + 1.02e4i)43-s + (−2.98e3 + 5.17e3i)47-s + ⋯
L(s)  = 1  + (−0.515 − 0.892i)5-s + (0.728 − 1.26i)7-s + (0.877 − 1.51i)11-s + (−0.652 − 1.13i)13-s + 0.151·17-s − 0.420·19-s + (0.715 + 1.23i)23-s + (−0.0313 + 0.0543i)25-s + (0.885 − 1.53i)29-s + (0.280 + 0.486i)31-s − 1.50·35-s − 0.182·37-s + (0.160 + 0.278i)41-s + (−0.487 + 0.843i)43-s + (−0.197 + 0.341i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.939 + 0.342i$
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: \((2,\ 324,\ (\ :5/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.799081049\)
\(L(\frac12)\) \(\approx\) \(1.799081049\)
\(L(\frac{7}{2})\) 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 \( 1 + (28.8 + 49.9i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-94.4 + 163. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-352. + 609. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (397. + 688. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 180.T + 1.41e6T^{2} \)
19 \( 1 + 661.T + 2.47e6T^{2} \)
23 \( 1 + (-1.81e3 - 3.14e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-4.01e3 + 6.94e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-1.50e3 - 2.60e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + 1.52e3T + 6.93e7T^{2} \)
41 \( 1 + (-1.73e3 - 2.99e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (5.90e3 - 1.02e4i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (2.98e3 - 5.17e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + 1.38e4T + 4.18e8T^{2} \)
59 \( 1 + (-1.13e4 - 1.96e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.87e4 + 3.23e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (3.54e4 + 6.14e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 6.77e4T + 1.80e9T^{2} \)
73 \( 1 + 3.15e4T + 2.07e9T^{2} \)
79 \( 1 + (3.13e4 - 5.43e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-4.67e4 + 8.09e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 6.87e4T + 5.58e9T^{2} \)
97 \( 1 + (5.88e4 - 1.01e5i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−10.50760590204143860511185511455, −9.370884177138578105952978337655, −8.193307257110001117127286578241, −7.85077398275131871907816732645, −6.47973142587112535973449269040, −5.19392906107106133389492475500, −4.27883397078509725698419031621, −3.24196430658223891223909671592, −1.16287401659765173976159738085, −0.53533899997943058342716654714, 1.74646894686509445662548768686, 2.67196272512244012045523504008, 4.22722083924676187958985279725, 5.09299147059005748163812636944, 6.68662005026930265593450348038, 7.10569293009904404911853892474, 8.479323806689972726683572745264, 9.232467093937306929177041205208, 10.31183083531718834659389500341, 11.36234482852995076819716223660

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