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Properties

Label 2-460-92.91-c1-0-12
Degree $2$
Conductor $460$
Sign $0.812 + 0.583i$
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  + (−0.265 − 1.38i)2-s − 0.0612i·3-s + (−1.85 + 0.737i)4-s + i·5-s + (−0.0850 + 0.0162i)6-s − 0.810·7-s + (1.51 + 2.38i)8-s + 2.99·9-s + (1.38 − 0.265i)10-s − 2.10·11-s + (0.0451 + 0.113i)12-s + 2.93·13-s + (0.215 + 1.12i)14-s + 0.0612·15-s + (2.91 − 2.74i)16-s + 4.65i·17-s + ⋯
L(s)  = 1  + (−0.187 − 0.982i)2-s − 0.0353i·3-s + (−0.929 + 0.368i)4-s + 0.447i·5-s + (−0.0347 + 0.00663i)6-s − 0.306·7-s + (0.536 + 0.843i)8-s + 0.998·9-s + (0.439 − 0.0838i)10-s − 0.633·11-s + (0.0130 + 0.0328i)12-s + 0.814·13-s + (0.0574 + 0.301i)14-s + 0.0158·15-s + (0.728 − 0.685i)16-s + 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.812 + 0.583i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.812 + 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17049 - 0.376503i\)
\(L(\frac12)\) \(\approx\) \(1.17049 - 0.376503i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.265 + 1.38i)T \)
5 \( 1 - iT \)
23 \( 1 + (-4.03 + 2.59i)T \)
good3 \( 1 + 0.0612iT - 3T^{2} \)
7 \( 1 + 0.810T + 7T^{2} \)
11 \( 1 + 2.10T + 11T^{2} \)
13 \( 1 - 2.93T + 13T^{2} \)
17 \( 1 - 4.65iT - 17T^{2} \)
19 \( 1 - 6.20T + 19T^{2} \)
29 \( 1 - 5.24T + 29T^{2} \)
31 \( 1 + 2.58iT - 31T^{2} \)
37 \( 1 - 2.99iT - 37T^{2} \)
41 \( 1 - 8.21T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 + 1.89iT - 47T^{2} \)
53 \( 1 - 8.12iT - 53T^{2} \)
59 \( 1 - 13.1iT - 59T^{2} \)
61 \( 1 + 13.7iT - 61T^{2} \)
67 \( 1 - 1.42T + 67T^{2} \)
71 \( 1 - 2.26iT - 71T^{2} \)
73 \( 1 + 4.13T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 + 8.73T + 83T^{2} \)
89 \( 1 + 5.82iT - 89T^{2} \)
97 \( 1 + 2.73iT - 97T^{2} \)
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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

−10.80598195217232005446313561769, −10.24381445424555626802090763894, −9.455300364143450914034286261359, −8.365966652616469907780985868809, −7.49898395359290323512437948048, −6.30669604253359202587465318801, −4.96877993887305450983169809771, −3.82950017695992429698121672995, −2.83713156862426828571034575997, −1.30552971756283034367686624868, 1.06726062452361544607458848497, 3.38091105349331994870739327899, 4.72881257974535400125454354572, 5.41492611648461168581341565613, 6.67691828985568157442388606891, 7.42005637937314267802603369159, 8.309580702622867806992367748345, 9.390527236099553915592277144742, 9.847098420096284552363843443167, 10.97852896851565101860747736778

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