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Properties

Label 2-18e2-12.11-c3-0-10
Degree $2$
Conductor $324$
Sign $-0.916 - 0.400i$
Analytic cond. $19.1166$
Root an. cond. $4.37225$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.36 + 1.54i)2-s + (3.20 − 7.32i)4-s − 1.43i·5-s + 27.5i·7-s + (3.76 + 22.3i)8-s + (2.21 + 3.38i)10-s − 22.2·11-s + 69.1·13-s + (−42.6 − 65.2i)14-s + (−43.4 − 46.9i)16-s − 31.4i·17-s − 11.4i·19-s + (−10.4 − 4.58i)20-s + (52.5 − 34.3i)22-s − 145.·23-s + ⋯
L(s)  = 1  + (−0.836 + 0.547i)2-s + (0.400 − 0.916i)4-s − 0.127i·5-s + 1.48i·7-s + (0.166 + 0.986i)8-s + (0.0700 + 0.107i)10-s − 0.608·11-s + 1.47·13-s + (−0.815 − 1.24i)14-s + (−0.678 − 0.734i)16-s − 0.448i·17-s − 0.138i·19-s + (−0.117 − 0.0512i)20-s + (0.509 − 0.333i)22-s − 1.31·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.916 - 0.400i$
Analytic conductor: \(19.1166\)
Root analytic conductor: \(4.37225\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ -0.916 - 0.400i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7460093069\)
\(L(\frac12)\) \(\approx\) \(0.7460093069\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.36 - 1.54i)T \)
3 \( 1 \)
good5 \( 1 + 1.43iT - 125T^{2} \)
7 \( 1 - 27.5iT - 343T^{2} \)
11 \( 1 + 22.2T + 1.33e3T^{2} \)
13 \( 1 - 69.1T + 2.19e3T^{2} \)
17 \( 1 + 31.4iT - 4.91e3T^{2} \)
19 \( 1 + 11.4iT - 6.85e3T^{2} \)
23 \( 1 + 145.T + 1.21e4T^{2} \)
29 \( 1 - 108. iT - 2.43e4T^{2} \)
31 \( 1 - 118. iT - 2.97e4T^{2} \)
37 \( 1 + 300.T + 5.06e4T^{2} \)
41 \( 1 - 398. iT - 6.89e4T^{2} \)
43 \( 1 - 200. iT - 7.95e4T^{2} \)
47 \( 1 + 303.T + 1.03e5T^{2} \)
53 \( 1 + 243. iT - 1.48e5T^{2} \)
59 \( 1 + 83.8T + 2.05e5T^{2} \)
61 \( 1 + 398.T + 2.26e5T^{2} \)
67 \( 1 - 355. iT - 3.00e5T^{2} \)
71 \( 1 + 866.T + 3.57e5T^{2} \)
73 \( 1 - 64.6T + 3.89e5T^{2} \)
79 \( 1 - 409. iT - 4.93e5T^{2} \)
83 \( 1 - 159.T + 5.71e5T^{2} \)
89 \( 1 - 1.49e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.40e3T + 9.12e5T^{2} \)
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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

−11.40919601472740071533446301446, −10.55507787030670463871774180260, −9.480125677525100230238381958020, −8.627678579978129587618404546629, −8.146053794098895342587393761783, −6.71396312203336429107662503525, −5.85396545822620256505943603504, −4.99201185224609487651616311192, −2.93354666793858519632947600824, −1.53742682684585580828861159324, 0.36276320830430111859125400165, 1.69672375206929340794888662240, 3.40138023840560178531481776802, 4.21577123986590555736787175522, 6.11663508211316209996343474024, 7.19702127970954985980754288934, 8.011916263513250718491784824840, 8.876878759323413694942391005543, 10.26091067514968392739030422825, 10.50433842444993658450773577420

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