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Properties

Label 2-456-57.32-c1-0-11
Degree $2$
Conductor $456$
Sign $0.973 - 0.228i$
Analytic cond. $3.64117$
Root an. cond. $1.90818$
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  + (1.68 + 0.405i)3-s + (3.00 − 0.529i)5-s + (−0.251 + 0.435i)7-s + (2.67 + 1.36i)9-s + (−2.58 + 1.49i)11-s + (0.364 + 1.00i)13-s + (5.27 + 0.325i)15-s + (−0.866 − 1.03i)17-s + (1.71 − 4.00i)19-s + (−0.599 + 0.630i)21-s + (−8.83 − 1.55i)23-s + (4.03 − 1.46i)25-s + (3.94 + 3.38i)27-s + (−3.43 − 2.88i)29-s + (3.58 + 2.06i)31-s + ⋯
L(s)  = 1  + (0.972 + 0.234i)3-s + (1.34 − 0.236i)5-s + (−0.0949 + 0.164i)7-s + (0.890 + 0.455i)9-s + (−0.779 + 0.450i)11-s + (0.101 + 0.277i)13-s + (1.36 + 0.0841i)15-s + (−0.210 − 0.250i)17-s + (0.394 − 0.918i)19-s + (−0.130 + 0.137i)21-s + (−1.84 − 0.324i)23-s + (0.807 − 0.293i)25-s + (0.759 + 0.650i)27-s + (−0.638 − 0.535i)29-s + (0.643 + 0.371i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.973 - 0.228i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ 0.973 - 0.228i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23687 + 0.258782i\)
\(L(\frac12)\) \(\approx\) \(2.23687 + 0.258782i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.68 - 0.405i)T \)
19 \( 1 + (-1.71 + 4.00i)T \)
good5 \( 1 + (-3.00 + 0.529i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (0.251 - 0.435i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.58 - 1.49i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.364 - 1.00i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.866 + 1.03i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (8.83 + 1.55i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (3.43 + 2.88i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-3.58 - 2.06i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.40iT - 37T^{2} \)
41 \( 1 + (-4.19 - 1.52i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.539 + 3.05i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (1.56 - 1.86i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (-1.97 + 11.2i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (8.56 - 7.18i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.52 + 8.62i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (6.94 - 8.27i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.02 + 11.5i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-9.45 - 3.44i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-0.310 + 0.852i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-6.19 - 3.57i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (13.7 - 5.00i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (9.45 + 11.2i)T + (-16.8 + 95.5i)T^{2} \)
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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

−10.80737404679403420891031664247, −9.852036528542061666993384799262, −9.514949680733792061052318077956, −8.527092777340032753798995592432, −7.58457431318711602583238377962, −6.42562220195972361574470834408, −5.33217501362587062445255386535, −4.30476730017411846095280290840, −2.73391381132640820883294220360, −1.91105416941547272722669855292, 1.71377492627439386721937539311, 2.74304090983598166146421506508, 3.94537521571522597041431989991, 5.57446554585855596318021615196, 6.27838142496728374209534150971, 7.55030293241044669093450498470, 8.272704877530779912164585142264, 9.366895351123366540321271343700, 10.02869087034079355618034385464, 10.67414866115214027305349885925

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