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Properties

Label 2-285-19.7-c1-0-7
Degree $2$
Conductor $285$
Sign $0.990 + 0.135i$
Analytic cond. $2.27573$
Root an. cond. $1.50855$
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  + (0.145 + 0.251i)2-s + (0.5 + 0.866i)3-s + (0.957 − 1.65i)4-s + (−0.5 − 0.866i)5-s + (−0.145 + 0.251i)6-s − 0.486·7-s + 1.13·8-s + (−0.499 + 0.866i)9-s + (0.145 − 0.251i)10-s + 5.34·11-s + 1.91·12-s + (1.24 − 2.15i)13-s + (−0.0707 − 0.122i)14-s + (0.499 − 0.866i)15-s + (−1.75 − 3.03i)16-s + (1.70 + 2.95i)17-s + ⋯
L(s)  = 1  + (0.102 + 0.178i)2-s + (0.288 + 0.499i)3-s + (0.478 − 0.829i)4-s + (−0.223 − 0.387i)5-s + (−0.0593 + 0.102i)6-s − 0.183·7-s + 0.402·8-s + (−0.166 + 0.288i)9-s + (0.0459 − 0.0796i)10-s + 1.61·11-s + 0.552·12-s + (0.344 − 0.597i)13-s + (−0.0189 − 0.0327i)14-s + (0.129 − 0.223i)15-s + (−0.437 − 0.757i)16-s + (0.414 + 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.990 + 0.135i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ 0.990 + 0.135i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63745 - 0.111505i\)
\(L(\frac12)\) \(\approx\) \(1.63745 - 0.111505i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (2.46 + 3.59i)T \)
good2 \( 1 + (-0.145 - 0.251i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + 0.486T + 7T^{2} \)
11 \( 1 - 5.34T + 11T^{2} \)
13 \( 1 + (-1.24 + 2.15i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.70 - 2.95i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.20 - 5.55i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.38 - 2.39i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.83T + 31T^{2} \)
37 \( 1 + 6.31T + 37T^{2} \)
41 \( 1 + (1.29 + 2.23i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.24 - 2.15i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.55 - 9.62i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.49 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.36 + 11.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.92 - 10.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.58 + 13.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.96 - 8.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.50 - 9.53i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.06 + 7.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.86T + 83T^{2} \)
89 \( 1 + (-3.25 + 5.63i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.63340475303258546758690297190, −10.87663417907933697673648070505, −9.843626590224253835454608729052, −9.125301965906191822633858934682, −8.017714078729753409872998413831, −6.71016893453626502081699478718, −5.84355113296785862686941600461, −4.62864474495695270561535133930, −3.44722866054428409408213419904, −1.51178953260330368712617651476, 1.89872647627973849438899158514, 3.33026795117625604315050708307, 4.21647578514445554396164801242, 6.37284588679574930833195525520, 6.82188613001730482571977090300, 7.993826833989961447789422332709, 8.786856822291465021310465036021, 9.989995273287482653751368739898, 11.25426175591520537954082720505, 11.96849321464538048124685939905

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