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Properties

Label 2-18e2-36.11-c1-0-5
Degree $2$
Conductor $324$
Sign $-0.573 - 0.819i$
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  + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (−3.79 + 2.19i)5-s + 2.82i·8-s − 6.19·10-s + (1.59 + 2.76i)13-s + (−2.00 + 3.46i)16-s − 0.138i·17-s + (−7.58 − 4.38i)20-s + (7.09 − 12.2i)25-s + 4.52i·26-s + (7.46 + 4.31i)29-s + (−4.89 + 2.82i)32-s + (0.0980 − 0.169i)34-s + 9.39·37-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (0.499 + 0.866i)4-s + (−1.69 + 0.979i)5-s + 0.999i·8-s − 1.95·10-s + (0.443 + 0.767i)13-s + (−0.500 + 0.866i)16-s − 0.0336i·17-s + (−1.69 − 0.979i)20-s + (1.41 − 2.45i)25-s + 0.886i·26-s + (1.38 + 0.800i)29-s + (−0.866 + 0.499i)32-s + (0.0168 − 0.0291i)34-s + 1.54·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.573 - 0.819i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ -0.573 - 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.707352 + 1.35881i\)
\(L(\frac12)\) \(\approx\) \(0.707352 + 1.35881i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 \)
good5 \( 1 + (3.79 - 2.19i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.59 - 2.76i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.138iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.46 - 4.31i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.39T + 37T^{2} \)
41 \( 1 + (-1.22 + 0.707i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.07iT - 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.69 - 13.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2.80T + 73T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.8iT - 89T^{2} \)
97 \( 1 + (4 - 6.92i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.81879355565316173738909137065, −11.38794804114624665713593816444, −10.47465998601171563814506599382, −8.703301429309538194507693245070, −7.87208237068849376914128194443, −7.03237758286051682911321635416, −6.31429671252629723797484124059, −4.66938129933162497760007815262, −3.83498176634655985044399833025, −2.84204931387701837945935239171, 0.883837178750420943295331817178, 3.08241456519540723097113642259, 4.16376931893620988673895029856, 4.88578337414475415311172885628, 6.16529693611194098725336287595, 7.56180787493913161552309677502, 8.318557259669291896503418558470, 9.509186552847065374212389324624, 10.79891015928405408111255885876, 11.46225367675752596958890642790

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