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Properties

Label 2-460-23.18-c1-0-5
Degree $2$
Conductor $460$
Sign $0.359 + 0.932i$
Analytic cond. $3.67311$
Root an. cond. $1.91653$
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  + (1.89 − 2.18i)3-s + (0.142 − 0.989i)5-s + (1.51 + 3.32i)7-s + (−0.766 − 5.33i)9-s + (0.489 − 0.143i)11-s + (2.48 − 5.44i)13-s + (−1.89 − 2.18i)15-s + (−3.94 + 2.53i)17-s + (6.00 + 3.85i)19-s + (10.1 + 2.98i)21-s + (−4.18 − 2.34i)23-s + (−0.959 − 0.281i)25-s + (−5.81 − 3.73i)27-s + (−5.84 + 3.75i)29-s + (−6.23 − 7.19i)31-s + ⋯
L(s)  = 1  + (1.09 − 1.26i)3-s + (0.0636 − 0.442i)5-s + (0.573 + 1.25i)7-s + (−0.255 − 1.77i)9-s + (0.147 − 0.0433i)11-s + (0.690 − 1.51i)13-s + (−0.489 − 0.565i)15-s + (−0.955 + 0.614i)17-s + (1.37 + 0.884i)19-s + (2.21 + 0.650i)21-s + (−0.871 − 0.489i)23-s + (−0.191 − 0.0563i)25-s + (−1.11 − 0.719i)27-s + (−1.08 + 0.697i)29-s + (−1.12 − 1.29i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.359 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.359 + 0.932i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.359 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73355 - 1.18935i\)
\(L(\frac12)\) \(\approx\) \(1.73355 - 1.18935i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.142 + 0.989i)T \)
23 \( 1 + (4.18 + 2.34i)T \)
good3 \( 1 + (-1.89 + 2.18i)T + (-0.426 - 2.96i)T^{2} \)
7 \( 1 + (-1.51 - 3.32i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (-0.489 + 0.143i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-2.48 + 5.44i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (3.94 - 2.53i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-6.00 - 3.85i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (5.84 - 3.75i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (6.23 + 7.19i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (0.102 + 0.713i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (1.77 - 12.3i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (2.54 - 2.93i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 - 8.44T + 47T^{2} \)
53 \( 1 + (-1.80 - 3.95i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (-1.29 + 2.82i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (-0.960 - 1.10i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (1.34 + 0.394i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (2.01 + 0.592i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (-13.4 - 8.65i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (-0.881 + 1.93i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-0.539 - 3.75i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (5.21 - 6.01i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (-1.22 + 8.53i)T + (-93.0 - 27.3i)T^{2} \)
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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

−11.10307596853593829074544351761, −9.643308453577847655849349651214, −8.763073488480707825515330070712, −8.165208939630717344213886348974, −7.60176581879539212775955563359, −6.15330056389933990120817331342, −5.46267514941301459822858615259, −3.65063149963782602422193432544, −2.45551277563662190505082008724, −1.43250727634699420766834721927, 2.04329651090817836604315083305, 3.60460417602063228274435787191, 4.10301639316449694866912979127, 5.16110096553891093219887104453, 6.92432316170467322445814198179, 7.56637499182181729343062238953, 8.897373825295578286033738073476, 9.293674407417671808658170692282, 10.30185575700680472667731114410, 11.02264252305231065562268542604

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