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Properties

Label 2-3e2-3.2-c18-0-1
Degree $2$
Conductor $9$
Sign $-0.577 + 0.816i$
Analytic cond. $18.4847$
Root an. cond. $4.29938$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 716. i·2-s − 2.51e5·4-s + 3.23e6i·5-s + 1.43e7·7-s + 7.64e6i·8-s − 2.31e9·10-s + 3.07e9i·11-s − 7.66e9·13-s + 1.02e10i·14-s − 7.14e10·16-s − 1.87e11i·17-s + 5.61e11·19-s − 8.13e11i·20-s − 2.20e12·22-s − 1.32e12i·23-s + ⋯
L(s)  = 1  + 1.39i·2-s − 0.959·4-s + 1.65i·5-s + 0.354·7-s + 0.0569i·8-s − 2.31·10-s + 1.30i·11-s − 0.723·13-s + 0.496i·14-s − 1.03·16-s − 1.58i·17-s + 1.73·19-s − 1.58i·20-s − 1.82·22-s − 0.733i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(18.4847\)
Root analytic conductor: \(4.29938\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{9} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :9),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.697562 - 1.34758i\)
\(L(\frac12)\) \(\approx\) \(0.697562 - 1.34758i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 - 716. iT - 2.62e5T^{2} \)
5 \( 1 - 3.23e6iT - 3.81e12T^{2} \)
7 \( 1 - 1.43e7T + 1.62e15T^{2} \)
11 \( 1 - 3.07e9iT - 5.55e18T^{2} \)
13 \( 1 + 7.66e9T + 1.12e20T^{2} \)
17 \( 1 + 1.87e11iT - 1.40e22T^{2} \)
19 \( 1 - 5.61e11T + 1.04e23T^{2} \)
23 \( 1 + 1.32e12iT - 3.24e24T^{2} \)
29 \( 1 - 7.53e12iT - 2.10e26T^{2} \)
31 \( 1 - 1.19e12T + 6.99e26T^{2} \)
37 \( 1 + 7.69e13T + 1.68e28T^{2} \)
41 \( 1 - 1.99e14iT - 1.07e29T^{2} \)
43 \( 1 + 1.24e14T + 2.52e29T^{2} \)
47 \( 1 - 7.45e14iT - 1.25e30T^{2} \)
53 \( 1 + 3.21e14iT - 1.08e31T^{2} \)
59 \( 1 - 1.41e16iT - 7.50e31T^{2} \)
61 \( 1 - 2.06e16T + 1.36e32T^{2} \)
67 \( 1 - 2.44e16T + 7.40e32T^{2} \)
71 \( 1 - 2.96e16iT - 2.10e33T^{2} \)
73 \( 1 - 3.70e16T + 3.46e33T^{2} \)
79 \( 1 + 9.35e16T + 1.43e34T^{2} \)
83 \( 1 + 4.70e16iT - 3.49e34T^{2} \)
89 \( 1 + 1.38e17iT - 1.22e35T^{2} \)
97 \( 1 + 1.61e17T + 5.77e35T^{2} \)
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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

−17.69516095782873098009438111365, −15.95333321192238158579135854226, −14.78453627929658910323548670580, −14.10546917847517602661780049233, −11.57748886520543884671596612604, −9.789509460929665850648694985645, −7.47631599893058164614049092141, −6.89740048230180916161208932882, −5.04646788298359058574337499328, −2.62074377905480280187360243857, 0.57738083898090725159036934247, 1.63361011555887291374086900046, 3.65527566101630883096175210232, 5.27752533573388415204182519125, 8.349634369808378399561959331631, 9.690595715415190087998231799356, 11.41069761020371683718346392891, 12.48376223296660648122445443820, 13.62667387341056896591121141491, 16.00753724679412050556095468509

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