Properties

Label 2-18e2-9.7-c3-0-3
Degree $2$
Conductor $324$
Sign $0.939 - 0.342i$
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  + (−4.5 − 7.79i)5-s + (0.5 − 0.866i)7-s + (−31.5 + 54.5i)11-s + (14 + 24.2i)13-s + 72·17-s + 98·19-s + (−63 − 109. i)23-s + (22 − 38.1i)25-s + (63 − 109. i)29-s + (129.5 + 224. i)31-s − 9·35-s + 386·37-s + (225 + 389. i)41-s + (17 − 29.4i)43-s + (27 − 46.7i)47-s + ⋯
L(s)  = 1  + (−0.402 − 0.697i)5-s + (0.0269 − 0.0467i)7-s + (−0.863 + 1.49i)11-s + (0.298 + 0.517i)13-s + 1.02·17-s + 1.18·19-s + (−0.571 − 0.989i)23-s + (0.175 − 0.304i)25-s + (0.403 − 0.698i)29-s + (0.750 + 1.29i)31-s − 0.0434·35-s + 1.71·37-s + (0.857 + 1.48i)41-s + (0.0602 − 0.104i)43-s + (0.0837 − 0.145i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(19.1166\)
Root analytic conductor: \(4.37225\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.645823225\)
\(L(\frac12)\) \(\approx\) \(1.645823225\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (4.5 + 7.79i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (31.5 - 54.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-14 - 24.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 72T + 4.91e3T^{2} \)
19 \( 1 - 98T + 6.85e3T^{2} \)
23 \( 1 + (63 + 109. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-63 + 109. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-129.5 - 224. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 386T + 5.06e4T^{2} \)
41 \( 1 + (-225 - 389. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-17 + 29.4i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-27 + 46.7i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 693T + 1.48e5T^{2} \)
59 \( 1 + (90 + 155. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-140 + 242. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-293 - 507. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 504T + 3.57e5T^{2} \)
73 \( 1 - 161T + 3.89e5T^{2} \)
79 \( 1 + (220 - 381. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (499.5 - 865. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 882T + 7.04e5T^{2} \)
97 \( 1 + (-360.5 + 624. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.35101465547225160168113612762, −10.10994286260179528909852532421, −9.536542371664218804918650587077, −8.200476841792196676564309207664, −7.63574343568576525627688197148, −6.37967877519881137023231582662, −5.00833592551767471313979028454, −4.31940513893412940706157992357, −2.68655719901948491506880040306, −1.06383263312562327872496585481, 0.77402054066306260397507537847, 2.87388971897964494457200464816, 3.60537207784656534884930927804, 5.35994749583048097088568535573, 6.07905970095647360429010729322, 7.59525428254638429934892376938, 7.982510118586333691003602260837, 9.317981175865497587191162646416, 10.35843789578670937672571032185, 11.14535436853415931281237418873

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