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Properties

Label 2-46-23.22-c10-0-11
Degree $2$
Conductor $46$
Sign $0.565 + 0.824i$
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 + 29.9·3-s + 512.·4-s + 5.62e3i·5-s − 678.·6-s − 2.38e4i·7-s − 1.15e4·8-s − 5.81e4·9-s − 1.27e5i·10-s − 4.33e4i·11-s + 1.53e4·12-s + 1.95e4·13-s + 5.39e5i·14-s + 1.68e5i·15-s + 2.62e5·16-s + 1.26e5i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.123·3-s + 0.500·4-s + 1.80i·5-s − 0.0872·6-s − 1.41i·7-s − 0.353·8-s − 0.984·9-s − 1.27i·10-s − 0.268i·11-s + 0.0616·12-s + 0.0526·13-s + 1.00i·14-s + 0.222i·15-s + 0.250·16-s + 0.0893i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46\)    =    \(2 \cdot 23\)
Sign: $0.565 + 0.824i$
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: \((2,\ 46,\ (\ :5),\ 0.565 + 0.824i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.835231 - 0.440242i\)
\(L(\frac12)\) \(\approx\) \(0.835231 - 0.440242i\)
\(L(6)\) 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 \( 1 + (3.63e6 + 5.30e6i)T \)
good3 \( 1 - 29.9T + 5.90e4T^{2} \)
5 \( 1 - 5.62e3iT - 9.76e6T^{2} \)
7 \( 1 + 2.38e4iT - 2.82e8T^{2} \)
11 \( 1 + 4.33e4iT - 2.59e10T^{2} \)
13 \( 1 - 1.95e4T + 1.37e11T^{2} \)
17 \( 1 - 1.26e5iT - 2.01e12T^{2} \)
19 \( 1 - 1.18e6iT - 6.13e12T^{2} \)
29 \( 1 - 4.00e7T + 4.20e14T^{2} \)
31 \( 1 - 2.76e7T + 8.19e14T^{2} \)
37 \( 1 + 8.15e7iT - 4.80e15T^{2} \)
41 \( 1 + 1.04e8T + 1.34e16T^{2} \)
43 \( 1 + 2.28e8iT - 2.16e16T^{2} \)
47 \( 1 - 3.52e8T + 5.25e16T^{2} \)
53 \( 1 + 1.62e8iT - 1.74e17T^{2} \)
59 \( 1 - 5.40e8T + 5.11e17T^{2} \)
61 \( 1 + 1.14e9iT - 7.13e17T^{2} \)
67 \( 1 - 7.93e8iT - 1.82e18T^{2} \)
71 \( 1 - 2.30e9T + 3.25e18T^{2} \)
73 \( 1 + 2.95e9T + 4.29e18T^{2} \)
79 \( 1 + 1.38e9iT - 9.46e18T^{2} \)
83 \( 1 - 9.06e8iT - 1.55e19T^{2} \)
89 \( 1 + 7.93e9iT - 3.11e19T^{2} \)
97 \( 1 - 3.23e9iT - 7.37e19T^{2} \)
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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

−13.95863960614924939689635637696, −11.77400238815381087314988615225, −10.62135406355635641799014983714, −10.23742129336342157399371479075, −8.323898155000062467603176038477, −7.16898620482051810472405541485, −6.24542361275132366057225897997, −3.67268828897760045166858234642, −2.48732088814727947355911642612, −0.42750360646776697022665935101, 1.08637813767772570594408722785, 2.57953334039993628232331663271, 4.89433707652246009969382226207, 5.99305270810549089739619461098, 8.240022014610889243062400688775, 8.733549198163082763477437811800, 9.704568979920569536140365660716, 11.74501892357815361612181704450, 12.21745277037798430582881333042, 13.61898177850728447353582306128

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