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Properties

Label 2-3-3.2-c24-0-1
Degree $2$
Conductor $3$
Sign $-0.839 + 0.543i$
Analytic cond. $10.9490$
Root an. cond. $3.30892$
Motivic weight $24$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.51e3i·2-s + (−4.46e5 + 2.88e5i)3-s + 1.04e7·4-s + 3.68e8i·5-s + (−7.25e8 − 1.12e9i)6-s − 8.30e9·7-s + 6.84e10i·8-s + (1.15e11 − 2.57e11i)9-s − 9.24e11·10-s − 3.87e12i·11-s + (−4.67e12 + 3.02e12i)12-s − 3.48e13·13-s − 2.08e13i·14-s + (−1.06e14 − 1.64e14i)15-s + 3.82e12·16-s + 2.61e14i·17-s + ⋯
L(s)  = 1  + 0.613i·2-s + (−0.839 + 0.543i)3-s + 0.624·4-s + 1.50i·5-s + (−0.333 − 0.514i)6-s − 0.600·7-s + 0.995i·8-s + (0.409 − 0.912i)9-s − 0.924·10-s − 1.23i·11-s + (−0.523 + 0.339i)12-s − 1.49·13-s − 0.368i·14-s + (−0.819 − 1.26i)15-s + 0.0135·16-s + 0.449i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.839 + 0.543i$
Analytic conductor: \(10.9490\)
Root analytic conductor: \(3.30892\)
Motivic weight: \(24\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :12),\ -0.839 + 0.543i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.235552 - 0.797239i\)
\(L(\frac12)\) \(\approx\) \(0.235552 - 0.797239i\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.46e5 - 2.88e5i)T \)
good2 \( 1 - 2.51e3iT - 1.67e7T^{2} \)
5 \( 1 - 3.68e8iT - 5.96e16T^{2} \)
7 \( 1 + 8.30e9T + 1.91e20T^{2} \)
11 \( 1 + 3.87e12iT - 9.84e24T^{2} \)
13 \( 1 + 3.48e13T + 5.42e26T^{2} \)
17 \( 1 - 2.61e14iT - 3.39e29T^{2} \)
19 \( 1 - 7.24e14T + 4.89e30T^{2} \)
23 \( 1 - 1.12e16iT - 4.80e32T^{2} \)
29 \( 1 + 1.35e17iT - 1.25e35T^{2} \)
31 \( 1 - 2.21e17T + 6.20e35T^{2} \)
37 \( 1 + 1.03e19T + 4.33e37T^{2} \)
41 \( 1 - 1.85e19iT - 5.09e38T^{2} \)
43 \( 1 - 2.09e19T + 1.59e39T^{2} \)
47 \( 1 - 5.93e19iT - 1.35e40T^{2} \)
53 \( 1 - 1.46e20iT - 2.41e41T^{2} \)
59 \( 1 - 3.04e21iT - 3.16e42T^{2} \)
61 \( 1 - 3.55e21T + 7.04e42T^{2} \)
67 \( 1 + 2.84e21T + 6.69e43T^{2} \)
71 \( 1 - 2.39e22iT - 2.69e44T^{2} \)
73 \( 1 + 1.66e22T + 5.24e44T^{2} \)
79 \( 1 + 6.08e22T + 3.49e45T^{2} \)
83 \( 1 + 7.63e22iT - 1.14e46T^{2} \)
89 \( 1 + 1.24e23iT - 6.10e46T^{2} \)
97 \( 1 - 9.42e23T + 4.81e47T^{2} \)
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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

−21.61445227155807054671955074445, −19.22235875365080858445507652969, −17.32224286605742360039748284801, −15.88949061805608343862059392396, −14.60680643996933495508973406653, −11.58466459425104290781776732412, −10.29037774195527955284593870794, −7.07245650441216570917958217340, −5.87481593353681542930751611414, −3.02855356145617913127565593212, 0.39189162539001633464885667364, 1.93949047850805886881082440041, 4.94145017193832196632629452459, 7.11665161871161613171616554627, 9.890131449050827775582157960062, 12.10788183534108869917706994055, 12.68697853670343120713327880852, 16.02037027806004813820156377549, 17.25295209562531826941242748180, 19.44635431756050667261985337768

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