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Properties

Label 2-18e2-12.11-c3-0-25
Degree $2$
Conductor $324$
Sign $0.964 - 0.265i$
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  + (−2.80 + 0.379i)2-s + (7.71 − 2.12i)4-s − 6.12i·5-s + 4.70i·7-s + (−20.8 + 8.88i)8-s + (2.32 + 17.1i)10-s − 13.0·11-s + 38.9·13-s + (−1.78 − 13.1i)14-s + (54.9 − 32.7i)16-s + 30.4i·17-s + 132. i·19-s + (−13.0 − 47.2i)20-s + (36.5 − 4.95i)22-s − 56.7·23-s + ⋯
L(s)  = 1  + (−0.990 + 0.134i)2-s + (0.964 − 0.265i)4-s − 0.548i·5-s + 0.253i·7-s + (−0.919 + 0.392i)8-s + (0.0735 + 0.543i)10-s − 0.357·11-s + 0.830·13-s + (−0.0340 − 0.251i)14-s + (0.858 − 0.512i)16-s + 0.434i·17-s + 1.59i·19-s + (−0.145 − 0.528i)20-s + (0.354 − 0.0479i)22-s − 0.514·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(19.1166\)
Root analytic conductor: \(4.37225\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ 0.964 - 0.265i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.140919517\)
\(L(\frac12)\) \(\approx\) \(1.140919517\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.80 - 0.379i)T \)
3 \( 1 \)
good5 \( 1 + 6.12iT - 125T^{2} \)
7 \( 1 - 4.70iT - 343T^{2} \)
11 \( 1 + 13.0T + 1.33e3T^{2} \)
13 \( 1 - 38.9T + 2.19e3T^{2} \)
17 \( 1 - 30.4iT - 4.91e3T^{2} \)
19 \( 1 - 132. iT - 6.85e3T^{2} \)
23 \( 1 + 56.7T + 1.21e4T^{2} \)
29 \( 1 + 259. iT - 2.43e4T^{2} \)
31 \( 1 + 208. iT - 2.97e4T^{2} \)
37 \( 1 - 67.4T + 5.06e4T^{2} \)
41 \( 1 - 276. iT - 6.89e4T^{2} \)
43 \( 1 - 220. iT - 7.95e4T^{2} \)
47 \( 1 - 452.T + 1.03e5T^{2} \)
53 \( 1 - 422. iT - 1.48e5T^{2} \)
59 \( 1 - 666.T + 2.05e5T^{2} \)
61 \( 1 - 212.T + 2.26e5T^{2} \)
67 \( 1 + 697. iT - 3.00e5T^{2} \)
71 \( 1 - 479.T + 3.57e5T^{2} \)
73 \( 1 - 668.T + 3.89e5T^{2} \)
79 \( 1 - 1.12e3iT - 4.93e5T^{2} \)
83 \( 1 - 947.T + 5.71e5T^{2} \)
89 \( 1 - 63.5iT - 7.04e5T^{2} \)
97 \( 1 + 605.T + 9.12e5T^{2} \)
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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

−11.06902614672195408121251069854, −10.13624849420311485996065614890, −9.342883122391301518864101017071, −8.268789272668600720995356576584, −7.84102081539971035892661751607, −6.31264444544836341600087706969, −5.64483565371408203049369972853, −3.96125662834816246279957972086, −2.30631233588117856959739291681, −0.939140591055061821616308066947, 0.78123121404387396176898610015, 2.42628597266157834460094540350, 3.55104232393775285107947085479, 5.31422374985046638215144771626, 6.76254540405232827879747874286, 7.19426217517068570101465423124, 8.530838995583942076438908997377, 9.137170369487753207939384077979, 10.46816680850911041433070697598, 10.78821069695940042491705301569

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