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Properties

Label 2-18e2-324.203-c1-0-27
Degree $2$
Conductor $324$
Sign $0.847 + 0.531i$
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  + (−1.37 − 0.335i)2-s + (1.58 − 0.692i)3-s + (1.77 + 0.923i)4-s + (−0.0526 + 0.222i)5-s + (−2.41 + 0.417i)6-s + (0.907 + 0.675i)7-s + (−2.12 − 1.86i)8-s + (2.04 − 2.19i)9-s + (0.147 − 0.287i)10-s + (0.316 + 0.335i)11-s + (3.45 + 0.236i)12-s + (3.18 − 1.59i)13-s + (−1.01 − 1.23i)14-s + (0.0703 + 0.389i)15-s + (2.29 + 3.27i)16-s + (0.720 + 1.97i)17-s + ⋯
L(s)  = 1  + (−0.971 − 0.237i)2-s + (0.916 − 0.399i)3-s + (0.887 + 0.461i)4-s + (−0.0235 + 0.0994i)5-s + (−0.985 + 0.170i)6-s + (0.342 + 0.255i)7-s + (−0.752 − 0.659i)8-s + (0.680 − 0.732i)9-s + (0.0465 − 0.0909i)10-s + (0.0953 + 0.101i)11-s + (0.997 + 0.0683i)12-s + (0.882 − 0.443i)13-s + (−0.272 − 0.329i)14-s + (0.0181 + 0.100i)15-s + (0.573 + 0.818i)16-s + (0.174 + 0.480i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.847 + 0.531i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.847 + 0.531i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20302 - 0.346158i\)
\(L(\frac12)\) \(\approx\) \(1.20302 - 0.346158i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.335i)T \)
3 \( 1 + (-1.58 + 0.692i)T \)
good5 \( 1 + (0.0526 - 0.222i)T + (-4.46 - 2.24i)T^{2} \)
7 \( 1 + (-0.907 - 0.675i)T + (2.00 + 6.70i)T^{2} \)
11 \( 1 + (-0.316 - 0.335i)T + (-0.639 + 10.9i)T^{2} \)
13 \( 1 + (-3.18 + 1.59i)T + (7.76 - 10.4i)T^{2} \)
17 \( 1 + (-0.720 - 1.97i)T + (-13.0 + 10.9i)T^{2} \)
19 \( 1 + (0.473 - 1.30i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (3.29 + 4.42i)T + (-6.59 + 22.0i)T^{2} \)
29 \( 1 + (-0.595 - 0.906i)T + (-11.4 + 26.6i)T^{2} \)
31 \( 1 + (4.90 + 2.11i)T + (21.2 + 22.5i)T^{2} \)
37 \( 1 + (-2.82 - 2.37i)T + (6.42 + 36.4i)T^{2} \)
41 \( 1 + (-8.94 - 0.520i)T + (40.7 + 4.75i)T^{2} \)
43 \( 1 + (2.02 + 0.607i)T + (35.9 + 23.6i)T^{2} \)
47 \( 1 + (0.569 + 1.32i)T + (-32.2 + 34.1i)T^{2} \)
53 \( 1 + (7.44 + 4.30i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.51 - 4.78i)T + (-3.43 - 58.9i)T^{2} \)
61 \( 1 + (7.36 + 0.861i)T + (59.3 + 14.0i)T^{2} \)
67 \( 1 + (0.140 - 0.214i)T + (-26.5 - 61.5i)T^{2} \)
71 \( 1 + (2.78 - 15.7i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-1.53 - 8.68i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (13.1 - 0.764i)T + (78.4 - 9.17i)T^{2} \)
83 \( 1 + (-0.326 - 5.60i)T + (-82.4 + 9.63i)T^{2} \)
89 \( 1 + (10.2 - 1.80i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (4.74 - 1.12i)T + (86.6 - 43.5i)T^{2} \)
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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

−11.36456459546492777365604326823, −10.47784620235023886607067517289, −9.523699882609877862169813612310, −8.558277682855039587378769072796, −8.081013280159850381405328317333, −7.02282406284694849571408095479, −5.99773529177704369280879770681, −3.93352429740884799335609993080, −2.74236703832068724567130737839, −1.43310865188635655243734133915, 1.59904241209490549013465392135, 3.10637966437127588802220546538, 4.54496117765087538858178650018, 6.02323423572187020123251252603, 7.29658666982594349404685100183, 8.019463117957454882946299587397, 8.985565966606130978302839536523, 9.518374015249065997728532940717, 10.67213057173048461876108628300, 11.24645111188309932389425349536

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