Properties

Degree $2$
Conductor $324$
Sign $0.996 - 0.0820i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.99 − 0.164i)2-s + (15.9 − 1.31i)4-s + 29.7·5-s + 59.8i·7-s + (63.5 − 7.86i)8-s + (118. − 4.89i)10-s − 225. i·11-s + 171.·13-s + (9.84 + 239. i)14-s + (252. − 41.8i)16-s − 99.0·17-s + 169. i·19-s + (474. − 39.0i)20-s + (−37.0 − 901. i)22-s + 358. i·23-s + ⋯
L(s)  = 1  + (0.999 − 0.0410i)2-s + (0.996 − 0.0820i)4-s + 1.19·5-s + 1.22i·7-s + (0.992 − 0.122i)8-s + (1.18 − 0.0489i)10-s − 1.86i·11-s + 1.01·13-s + (0.0502 + 1.22i)14-s + (0.986 − 0.163i)16-s − 0.342·17-s + 0.468i·19-s + (1.18 − 0.0977i)20-s + (−0.0765 − 1.86i)22-s + 0.678i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0820i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.996 - 0.0820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.996 - 0.0820i$
Motivic weight: \(4\)
Character: $\chi_{324} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :2),\ 0.996 - 0.0820i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(5.168872825\)
\(L(\frac12)\) \(\approx\) \(5.168872825\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.99 + 0.164i)T \)
3 \( 1 \)
good5 \( 1 - 29.7T + 625T^{2} \)
7 \( 1 - 59.8iT - 2.40e3T^{2} \)
11 \( 1 + 225. iT - 1.46e4T^{2} \)
13 \( 1 - 171.T + 2.85e4T^{2} \)
17 \( 1 + 99.0T + 8.35e4T^{2} \)
19 \( 1 - 169. iT - 1.30e5T^{2} \)
23 \( 1 - 358. iT - 2.79e5T^{2} \)
29 \( 1 + 18.0T + 7.07e5T^{2} \)
31 \( 1 - 775. iT - 9.23e5T^{2} \)
37 \( 1 - 609.T + 1.87e6T^{2} \)
41 \( 1 + 413.T + 2.82e6T^{2} \)
43 \( 1 + 306. iT - 3.41e6T^{2} \)
47 \( 1 + 2.62e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.03e3T + 7.89e6T^{2} \)
59 \( 1 + 2.59e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.41e3T + 1.38e7T^{2} \)
67 \( 1 - 5.99e3iT - 2.01e7T^{2} \)
71 \( 1 - 1.23e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.06e3T + 2.83e7T^{2} \)
79 \( 1 + 1.88e3iT - 3.89e7T^{2} \)
83 \( 1 - 6.73e3iT - 4.74e7T^{2} \)
89 \( 1 - 9.43e3T + 6.27e7T^{2} \)
97 \( 1 + 1.45e4T + 8.85e7T^{2} \)
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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

−11.19877933605301921706040475455, −10.28832951119918101668681835979, −9.039394540660053274442028513610, −8.280081580909675247113200024879, −6.57057902289789188383665955026, −5.77177375843164739415827597050, −5.45131433545125833423760889186, −3.64573904693983639451066666302, −2.62278809319823929957930990519, −1.42920566023922544472385411656, 1.37888724315826875950808226265, 2.43204993606157893841068328517, 4.03906460624737755537032115167, 4.79039743916848134107574960725, 6.08367345518894178339948211339, 6.84855372768757151303988027018, 7.72490929712187160760920407755, 9.364303831400266105993453935471, 10.27001978187158181804365294937, 10.87005635459919666038534426203

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