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Properties

Label 2-18e2-9.4-c1-0-0
Degree $2$
Conductor $324$
Sign $-0.173 - 0.984i$
Analytic cond. $2.58715$
Root an. cond. $1.60846$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 2.59i)5-s + (−1 − 1.73i)7-s + (3 + 5.19i)11-s + (−2.5 + 4.33i)13-s − 3·17-s + 2·19-s + (−3 + 5.19i)23-s + (−2 − 3.46i)25-s + (−1.5 − 2.59i)29-s + (2 − 3.46i)31-s + 6·35-s + 5·37-s + (3 − 5.19i)41-s + (5 + 8.66i)43-s + (1.50 − 2.59i)49-s + ⋯
L(s)  = 1  + (−0.670 + 1.16i)5-s + (−0.377 − 0.654i)7-s + (0.904 + 1.56i)11-s + (−0.693 + 1.20i)13-s − 0.727·17-s + 0.458·19-s + (−0.625 + 1.08i)23-s + (−0.400 − 0.692i)25-s + (−0.278 − 0.482i)29-s + (0.359 − 0.622i)31-s + 1.01·35-s + 0.821·37-s + (0.468 − 0.811i)41-s + (0.762 + 1.32i)43-s + (0.214 − 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613887 + 0.731602i\)
\(L(\frac12)\) \(\approx\) \(0.613887 + 0.731602i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.66754041152565186880485481359, −11.15038007385762031401981620100, −9.816204895353720041342496643460, −9.475339924485780284252381223583, −7.67787536090893974604946198624, −7.10471115453247176040541143517, −6.41053879883303465068158051864, −4.49067354636861185391041141943, −3.76327665947738011522374768354, −2.17036299705848216660305075705, 0.69069172963674813462339747625, 2.92098375982074329815751560924, 4.20951292402653755898014360815, 5.38715854225702502993223469578, 6.30655856326879286692742163827, 7.79071996353110318851168208698, 8.655620202190949565058895699010, 9.155390014874397930741743264054, 10.49034803084128120866187889568, 11.56229037963041920354744486466

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