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Properties

Label 2-18e2-4.3-c2-0-22
Degree $2$
Conductor $324$
Sign $0.00651 - 0.999i$
Analytic cond. $8.82836$
Root an. cond. $2.97125$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.40 + 1.41i)2-s + (−0.0260 + 3.99i)4-s + 8.06·5-s + 4.50i·7-s + (−5.71 + 5.60i)8-s + (11.3 + 11.4i)10-s − 3.76i·11-s + 7.05·13-s + (−6.39 + 6.35i)14-s + (−15.9 − 0.208i)16-s + 0.517·17-s − 16.4i·19-s + (−0.210 + 32.2i)20-s + (5.33 − 5.30i)22-s + 31.9i·23-s + ⋯
L(s)  = 1  + (0.704 + 0.709i)2-s + (−0.00651 + 0.999i)4-s + 1.61·5-s + 0.643i·7-s + (−0.713 + 0.700i)8-s + (1.13 + 1.14i)10-s − 0.342i·11-s + 0.542·13-s + (−0.456 + 0.453i)14-s + (−0.999 − 0.0130i)16-s + 0.0304·17-s − 0.864i·19-s + (−0.0105 + 1.61i)20-s + (0.242 − 0.241i)22-s + 1.39i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00651 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.00651 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.00651 - 0.999i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ 0.00651 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.12493 + 2.11113i\)
\(L(\frac12)\) \(\approx\) \(2.12493 + 2.11113i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 1.41i)T \)
3 \( 1 \)
good5 \( 1 - 8.06T + 25T^{2} \)
7 \( 1 - 4.50iT - 49T^{2} \)
11 \( 1 + 3.76iT - 121T^{2} \)
13 \( 1 - 7.05T + 169T^{2} \)
17 \( 1 - 0.517T + 289T^{2} \)
19 \( 1 + 16.4iT - 361T^{2} \)
23 \( 1 - 31.9iT - 529T^{2} \)
29 \( 1 + 18.9T + 841T^{2} \)
31 \( 1 + 15.1iT - 961T^{2} \)
37 \( 1 - 0.592T + 1.36e3T^{2} \)
41 \( 1 + 24.7T + 1.68e3T^{2} \)
43 \( 1 + 32.1iT - 1.84e3T^{2} \)
47 \( 1 - 60.5iT - 2.20e3T^{2} \)
53 \( 1 + 0.664T + 2.80e3T^{2} \)
59 \( 1 + 35.3iT - 3.48e3T^{2} \)
61 \( 1 + 67.5T + 3.72e3T^{2} \)
67 \( 1 + 85.9iT - 4.48e3T^{2} \)
71 \( 1 + 56.4iT - 5.04e3T^{2} \)
73 \( 1 - 131.T + 5.32e3T^{2} \)
79 \( 1 + 146. iT - 6.24e3T^{2} \)
83 \( 1 + 100. iT - 6.88e3T^{2} \)
89 \( 1 + 25.8T + 7.92e3T^{2} \)
97 \( 1 - 96.5T + 9.40e3T^{2} \)
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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

−11.79071145818228405444401452557, −10.80254398036195679160168735169, −9.409399725840365677689207413637, −8.948302306694315931790278559116, −7.64537688563222600412995395425, −6.39121079957120861517789654476, −5.79095681226030837256875175006, −4.97374109173128524611572906204, −3.30802202263756542240535372543, −2.03804463753001354282295997573, 1.31440648896698477934391180333, 2.44580538296053995822145154045, 3.90142136721426197374423654344, 5.13818002842387965371229285850, 6.04817464698830346667533584353, 6.88265232197233144101264305604, 8.632261080846725265688392565358, 9.734435454850885453072519698532, 10.27248739412280841622257829929, 11.00506701395938606859954202893

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