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Properties

Label 2-46-23.22-c2-0-3
Degree $2$
Conductor $46$
Sign $0.995 + 0.0933i$
Analytic cond. $1.25340$
Root an. cond. $1.11955$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + 0.414·3-s + 2.00·4-s − 5.18i·5-s + 0.585·6-s + 12.5i·7-s + 2.82·8-s − 8.82·9-s − 7.33i·10-s − 12.5i·11-s + 0.828·12-s − 12.3·13-s + 17.7i·14-s − 2.14i·15-s + 4.00·16-s + 19.8i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.138·3-s + 0.500·4-s − 1.03i·5-s + 0.0976·6-s + 1.78i·7-s + 0.353·8-s − 0.980·9-s − 0.733i·10-s − 1.13i·11-s + 0.0690·12-s − 0.947·13-s + 1.26i·14-s − 0.143i·15-s + 0.250·16-s + 1.16i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0933i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46\)    =    \(2 \cdot 23\)
Sign: $0.995 + 0.0933i$
Analytic conductor: \(1.25340\)
Root analytic conductor: \(1.11955\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{46} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 46,\ (\ :1),\ 0.995 + 0.0933i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.55184 - 0.0726170i\)
\(L(\frac12)\) \(\approx\) \(1.55184 - 0.0726170i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
23 \( 1 + (-22.8 - 2.14i)T \)
good3 \( 1 - 0.414T + 9T^{2} \)
5 \( 1 + 5.18iT - 25T^{2} \)
7 \( 1 - 12.5iT - 49T^{2} \)
11 \( 1 + 12.5iT - 121T^{2} \)
13 \( 1 + 12.3T + 169T^{2} \)
17 \( 1 - 19.8iT - 289T^{2} \)
19 \( 1 + 14.6iT - 361T^{2} \)
29 \( 1 - 6.85T + 841T^{2} \)
31 \( 1 - 24.7T + 961T^{2} \)
37 \( 1 + 50.9iT - 1.36e3T^{2} \)
41 \( 1 + 3.48T + 1.68e3T^{2} \)
43 \( 1 - 45.7iT - 1.84e3T^{2} \)
47 \( 1 - 17.0T + 2.20e3T^{2} \)
53 \( 1 + 54.3iT - 2.80e3T^{2} \)
59 \( 1 + 65.7T + 3.48e3T^{2} \)
61 \( 1 - 50.0iT - 3.72e3T^{2} \)
67 \( 1 - 6.44iT - 4.48e3T^{2} \)
71 \( 1 - 61.2T + 5.04e3T^{2} \)
73 \( 1 + 121.T + 5.32e3T^{2} \)
79 \( 1 - 23.2iT - 6.24e3T^{2} \)
83 \( 1 - 114. iT - 6.88e3T^{2} \)
89 \( 1 + 125. iT - 7.92e3T^{2} \)
97 \( 1 + 36.2iT - 9.40e3T^{2} \)
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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

−15.36174243647356130462489014706, −14.47986095008220451557709886844, −13.05613354092748568931731254494, −12.21882591763392044179300952043, −11.23806067101778874586055297804, −9.085518435947728808356705113846, −8.375831873375352021526140708066, −5.99680452630911938524904863470, −5.08379865006594760243837393286, −2.75898084878820557646188964776, 3.00189942549558372280637509661, 4.69214458360028116478397518667, 6.79213840072646953812474322015, 7.53585964861087010064605321275, 9.926284373702534829339231768328, 10.84684826055563989559029151213, 12.07588944190213767284019764022, 13.64495961066523625528928912553, 14.29940601235138906964704433398, 15.13687845167655328585987862330

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