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Properties

Label 2-18e2-12.11-c1-0-17
Degree $2$
Conductor $324$
Sign $0.537 + 0.843i$
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  + (1.35 − 0.396i)2-s + (1.68 − 1.07i)4-s − 2.52i·5-s + 1.27i·7-s + (1.86 − 2.12i)8-s + (−1 − 3.42i)10-s + 0.505·11-s − 2.37·13-s + (0.505 + 1.73i)14-s + (1.68 − 3.62i)16-s − 0.792i·17-s + 4.70i·19-s + (−2.71 − 4.25i)20-s + (0.686 − 0.200i)22-s + 3.22·23-s + ⋯
L(s)  = 1  + (0.959 − 0.280i)2-s + (0.843 − 0.537i)4-s − 1.12i·5-s + 0.482i·7-s + (0.658 − 0.752i)8-s + (−0.316 − 1.08i)10-s + 0.152·11-s − 0.657·13-s + (0.135 + 0.462i)14-s + (0.421 − 0.906i)16-s − 0.192i·17-s + 1.07i·19-s + (−0.607 − 0.951i)20-s + (0.146 − 0.0426i)22-s + 0.671·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.537 + 0.843i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.537 + 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00376 - 1.09852i\)
\(L(\frac12)\) \(\approx\) \(2.00376 - 1.09852i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 + 0.396i)T \)
3 \( 1 \)
good5 \( 1 + 2.52iT - 5T^{2} \)
7 \( 1 - 1.27iT - 7T^{2} \)
11 \( 1 - 0.505T + 11T^{2} \)
13 \( 1 + 2.37T + 13T^{2} \)
17 \( 1 + 0.792iT - 17T^{2} \)
19 \( 1 - 4.70iT - 19T^{2} \)
23 \( 1 - 3.22T + 23T^{2} \)
29 \( 1 + 2.52iT - 29T^{2} \)
31 \( 1 - 8.12iT - 31T^{2} \)
37 \( 1 + 6.74T + 37T^{2} \)
41 \( 1 - 6.78iT - 41T^{2} \)
43 \( 1 - 7.72iT - 43T^{2} \)
47 \( 1 - 1.19T + 47T^{2} \)
53 \( 1 + 1.87iT - 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 + 2.37T + 61T^{2} \)
67 \( 1 + 7.72iT - 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 - 3.37T + 73T^{2} \)
79 \( 1 + 9.88iT - 79T^{2} \)
83 \( 1 + 7.64T + 83T^{2} \)
89 \( 1 - 11.9iT - 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−11.90470900169699834693738613690, −10.69894505825292609215203841678, −9.686466463982851975350633338596, −8.725417173456434199129307823174, −7.54386655144225961658513921803, −6.28712905220404262036802236624, −5.23515222780875294334058602536, −4.54193886022331215679694501071, −3.10651977497376665175931290415, −1.53111853418433834190089795416, 2.41528668247720303456925436152, 3.49744917356767410903429624991, 4.67753710928310399042654080518, 5.89779233465055837647805542090, 7.08347782761965572025354382950, 7.31056059999400564892911307684, 8.852229625231463395666676596518, 10.29828062091684687909161691814, 10.93790762447934821861362313377, 11.78020739086738426516913895874

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