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Properties

Label 2-46-23.22-c10-0-12
Degree $2$
Conductor $46$
Sign $0.877 - 0.479i$
Analytic cond. $29.2264$
Root an. cond. $5.40614$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 22.6·2-s + 374.·3-s + 512.·4-s + 4.30e3i·5-s + 8.46e3·6-s − 1.39e4i·7-s + 1.15e4·8-s + 8.09e4·9-s + 9.73e4i·10-s − 1.07e5i·11-s + 1.91e5·12-s + 5.27e5·13-s − 3.15e5i·14-s + 1.61e6i·15-s + 2.62e5·16-s + 1.20e6i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.54·3-s + 0.500·4-s + 1.37i·5-s + 1.08·6-s − 0.830i·7-s + 0.353·8-s + 1.37·9-s + 0.973i·10-s − 0.664i·11-s + 0.770·12-s + 1.42·13-s − 0.587i·14-s + 2.12i·15-s + 0.250·16-s + 0.847i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46\)    =    \(2 \cdot 23\)
Sign: $0.877 - 0.479i$
Analytic conductor: \(29.2264\)
Root analytic conductor: \(5.40614\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{46} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 46,\ (\ :5),\ 0.877 - 0.479i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(5.32263 + 1.35948i\)
\(L(\frac12)\) \(\approx\) \(5.32263 + 1.35948i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 22.6T \)
23 \( 1 + (-5.64e6 + 3.08e6i)T \)
good3 \( 1 - 374.T + 5.90e4T^{2} \)
5 \( 1 - 4.30e3iT - 9.76e6T^{2} \)
7 \( 1 + 1.39e4iT - 2.82e8T^{2} \)
11 \( 1 + 1.07e5iT - 2.59e10T^{2} \)
13 \( 1 - 5.27e5T + 1.37e11T^{2} \)
17 \( 1 - 1.20e6iT - 2.01e12T^{2} \)
19 \( 1 - 2.86e6iT - 6.13e12T^{2} \)
29 \( 1 + 2.41e7T + 4.20e14T^{2} \)
31 \( 1 + 2.35e7T + 8.19e14T^{2} \)
37 \( 1 - 1.08e8iT - 4.80e15T^{2} \)
41 \( 1 - 1.79e7T + 1.34e16T^{2} \)
43 \( 1 + 2.57e8iT - 2.16e16T^{2} \)
47 \( 1 + 1.61e8T + 5.25e16T^{2} \)
53 \( 1 + 2.62e8iT - 1.74e17T^{2} \)
59 \( 1 - 1.86e8T + 5.11e17T^{2} \)
61 \( 1 + 1.18e9iT - 7.13e17T^{2} \)
67 \( 1 + 2.19e9iT - 1.82e18T^{2} \)
71 \( 1 + 2.05e9T + 3.25e18T^{2} \)
73 \( 1 + 2.02e9T + 4.29e18T^{2} \)
79 \( 1 + 2.67e9iT - 9.46e18T^{2} \)
83 \( 1 + 3.82e7iT - 1.55e19T^{2} \)
89 \( 1 - 1.73e8iT - 3.11e19T^{2} \)
97 \( 1 + 3.89e9iT - 7.37e19T^{2} \)
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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

−13.85522909457841887683460226823, −13.11134233653826132177657642481, −11.11069061697833498736590736727, −10.29969636811669003664770648323, −8.558921213289243835300634627961, −7.42809455783902969373719974114, −6.23088923472240413001954433024, −3.68912837930076883430225553102, −3.33728236655912502718120576212, −1.75183452757511578572827261561, 1.40751753877367542478883848980, 2.72139704692774738373683934086, 4.13670220302166438999701348584, 5.44191026068349345980402510587, 7.44026605051045161735236580933, 8.852059944702319130094656104817, 9.227155143035230965549052041356, 11.43931604699770980077531619685, 12.95147077744496176072329936857, 13.23521572685516250454578962469

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