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Properties

Label 2-18e2-108.47-c1-0-11
Degree $2$
Conductor $324$
Sign $0.899 + 0.436i$
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  + (−1.32 + 0.504i)2-s + (1.49 − 1.33i)4-s + (3.39 − 0.598i)5-s + (1.88 − 2.24i)7-s + (−1.29 + 2.51i)8-s + (−4.18 + 2.50i)10-s + (0.453 − 2.57i)11-s + (−5.09 − 1.85i)13-s + (−1.35 + 3.91i)14-s + (0.447 − 3.97i)16-s + (−1.15 + 0.667i)17-s + (0.0790 + 0.0456i)19-s + (4.26 − 5.41i)20-s + (0.698 + 3.63i)22-s + (−0.650 + 0.545i)23-s + ⋯
L(s)  = 1  + (−0.934 + 0.356i)2-s + (0.745 − 0.666i)4-s + (1.51 − 0.267i)5-s + (0.711 − 0.848i)7-s + (−0.458 + 0.888i)8-s + (−1.32 + 0.791i)10-s + (0.136 − 0.776i)11-s + (−1.41 − 0.514i)13-s + (−0.362 + 1.04i)14-s + (0.111 − 0.993i)16-s + (−0.280 + 0.161i)17-s + (0.0181 + 0.0104i)19-s + (0.953 − 1.21i)20-s + (0.148 + 0.773i)22-s + (−0.135 + 0.113i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.899 + 0.436i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.899 + 0.436i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10312 - 0.253628i\)
\(L(\frac12)\) \(\approx\) \(1.10312 - 0.253628i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 - 0.504i)T \)
3 \( 1 \)
good5 \( 1 + (-3.39 + 0.598i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (-1.88 + 2.24i)T + (-1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.453 + 2.57i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (5.09 + 1.85i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.15 - 0.667i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0790 - 0.0456i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.650 - 0.545i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.171 - 0.470i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-4.28 - 5.10i)T + (-5.38 + 30.5i)T^{2} \)
37 \( 1 + (-2.75 - 4.77i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.976 + 2.68i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (-5.59 - 0.986i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (5.06 + 4.24i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 7.37iT - 53T^{2} \)
59 \( 1 + (-0.528 - 2.99i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (1.85 + 1.55i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-1.64 + 4.53i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-5.91 - 10.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.83 - 10.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.576 + 1.58i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (2.02 - 0.738i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-12.5 - 7.26i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.14 + 6.49i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.17007724762685090625540563521, −10.32601515347338495916469737522, −9.767214204727312888630707385478, −8.782638043085251066114371469633, −7.82920946104849776651741196929, −6.79838595098965390027356132774, −5.76960518765285149087702775534, −4.86081759640506033182298244522, −2.56664101803021246292137445189, −1.20806818802886660981418565260, 1.95433705755770500652634492098, 2.49496207117539482655236204618, 4.71582609970004332694078849796, 5.96963007715376803017256606916, 6.96222290371241574242391383917, 8.023972525410283612149802851715, 9.344965365203253225024431277183, 9.559076647479148974875875298654, 10.53716614898130018406757468689, 11.61402246911122454505830581567

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