Properties

Label 2-46-23.6-c3-0-0
Degree $2$
Conductor $46$
Sign $-0.968 - 0.248i$
Analytic cond. $2.71408$
Root an. cond. $1.64744$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.830 + 1.81i)2-s + (−0.430 − 0.126i)3-s + (−2.61 − 3.02i)4-s + (−13.5 + 8.73i)5-s + (0.587 − 0.678i)6-s + (−2.74 + 19.0i)7-s + (7.67 − 2.25i)8-s + (−22.5 − 14.4i)9-s + (−4.59 − 31.9i)10-s + (−12.4 − 27.1i)11-s + (0.745 + 1.63i)12-s + (12.6 + 88.0i)13-s + (−32.4 − 20.8i)14-s + (6.95 − 2.04i)15-s + (−2.27 + 15.8i)16-s + (61.4 − 70.9i)17-s + ⋯
L(s)  = 1  + (−0.293 + 0.643i)2-s + (−0.0828 − 0.0243i)3-s + (−0.327 − 0.377i)4-s + (−1.21 + 0.781i)5-s + (0.0399 − 0.0461i)6-s + (−0.147 + 1.02i)7-s + (0.339 − 0.0996i)8-s + (−0.834 − 0.536i)9-s + (−0.145 − 1.01i)10-s + (−0.340 − 0.745i)11-s + (0.0179 + 0.0392i)12-s + (0.270 + 1.87i)13-s + (−0.618 − 0.397i)14-s + (0.119 − 0.0351i)15-s + (−0.0355 + 0.247i)16-s + (0.877 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46\)    =    \(2 \cdot 23\)
Sign: $-0.968 - 0.248i$
Analytic conductor: \(2.71408\)
Root analytic conductor: \(1.64744\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{46} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 46,\ (\ :3/2),\ -0.968 - 0.248i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0690776 + 0.546352i\)
\(L(\frac12)\) \(\approx\) \(0.0690776 + 0.546352i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.830 - 1.81i)T \)
23 \( 1 + (14.7 - 109. i)T \)
good3 \( 1 + (0.430 + 0.126i)T + (22.7 + 14.5i)T^{2} \)
5 \( 1 + (13.5 - 8.73i)T + (51.9 - 113. i)T^{2} \)
7 \( 1 + (2.74 - 19.0i)T + (-329. - 96.6i)T^{2} \)
11 \( 1 + (12.4 + 27.1i)T + (-871. + 1.00e3i)T^{2} \)
13 \( 1 + (-12.6 - 88.0i)T + (-2.10e3 + 618. i)T^{2} \)
17 \( 1 + (-61.4 + 70.9i)T + (-699. - 4.86e3i)T^{2} \)
19 \( 1 + (-54.5 - 62.9i)T + (-976. + 6.78e3i)T^{2} \)
29 \( 1 + (-58.6 + 67.7i)T + (-3.47e3 - 2.41e4i)T^{2} \)
31 \( 1 + (227. - 66.8i)T + (2.50e4 - 1.61e4i)T^{2} \)
37 \( 1 + (-60.3 - 38.7i)T + (2.10e4 + 4.60e4i)T^{2} \)
41 \( 1 + (62.0 - 39.8i)T + (2.86e4 - 6.26e4i)T^{2} \)
43 \( 1 + (237. + 69.6i)T + (6.68e4 + 4.29e4i)T^{2} \)
47 \( 1 + 380.T + 1.03e5T^{2} \)
53 \( 1 + (-5.45 + 37.9i)T + (-1.42e5 - 4.19e4i)T^{2} \)
59 \( 1 + (-88.2 - 613. i)T + (-1.97e5 + 5.78e4i)T^{2} \)
61 \( 1 + (-597. + 175. i)T + (1.90e5 - 1.22e5i)T^{2} \)
67 \( 1 + (37.9 - 82.9i)T + (-1.96e5 - 2.27e5i)T^{2} \)
71 \( 1 + (136. - 299. i)T + (-2.34e5 - 2.70e5i)T^{2} \)
73 \( 1 + (-348. - 401. i)T + (-5.53e4 + 3.85e5i)T^{2} \)
79 \( 1 + (29.7 + 206. i)T + (-4.73e5 + 1.38e5i)T^{2} \)
83 \( 1 + (-746. - 480. i)T + (2.37e5 + 5.20e5i)T^{2} \)
89 \( 1 + (1.07e3 + 316. i)T + (5.93e5 + 3.81e5i)T^{2} \)
97 \( 1 + (-777. + 499. i)T + (3.79e5 - 8.30e5i)T^{2} \)
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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

−15.90818219819004807369532566133, −14.77833130626951300969664410425, −13.98914061802492788903709813143, −11.83265954933329964640336912855, −11.45959134511902758679296516234, −9.467021629432708667228930615186, −8.329704197278105034715896696289, −7.01342741966255303223269974427, −5.65592010116795517821557499590, −3.41267324187483906303572453384, 0.48309389412480203977240460292, 3.47788899464059827476107819113, 5.04373853594834093431826172574, 7.66513373350260570379100935348, 8.349509004440309786922979872448, 10.19682348713506387629895162885, 11.08920778093533851314160163596, 12.42645112121390655906042679167, 13.16410744508771725431771412979, 14.81151813615202331379129632809

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