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Properties

Label 2-285-19.11-c1-0-5
Degree $2$
Conductor $285$
Sign $0.636 + 0.771i$
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  + (0.809 − 1.40i)2-s + (−0.5 + 0.866i)3-s + (−0.309 − 0.535i)4-s + (0.5 − 0.866i)5-s + (0.809 + 1.40i)6-s + 0.763·7-s + 2.23·8-s + (−0.499 − 0.866i)9-s + (−0.809 − 1.40i)10-s + 2·11-s + 0.618·12-s + (0.618 + 1.07i)13-s + (0.618 − 1.07i)14-s + (0.499 + 0.866i)15-s + (2.42 − 4.20i)16-s + (1.73 − 3.00i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.990i)2-s + (−0.288 + 0.499i)3-s + (−0.154 − 0.267i)4-s + (0.223 − 0.387i)5-s + (0.330 + 0.572i)6-s + 0.288·7-s + 0.790·8-s + (−0.166 − 0.288i)9-s + (−0.255 − 0.443i)10-s + 0.603·11-s + 0.178·12-s + (0.171 + 0.296i)13-s + (0.165 − 0.286i)14-s + (0.129 + 0.223i)15-s + (0.606 − 1.05i)16-s + (0.421 − 0.729i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.636 + 0.771i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ 0.636 + 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62534 - 0.766434i\)
\(L(\frac12)\) \(\approx\) \(1.62534 - 0.766434i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.35 + 0.204i)T \)
good2 \( 1 + (-0.809 + 1.40i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 - 0.763T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (-0.618 - 1.07i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.73 + 3.00i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.23 - 2.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.381 - 0.661i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.23T + 31T^{2} \)
37 \( 1 - 2.47T + 37T^{2} \)
41 \( 1 + (3.85 - 6.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.85 - 10.1i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.881 + 1.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.23 + 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.61 - 2.80i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.61 - 7.99i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.76 + 3.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 + (1.76 + 3.05i)T + (-44.5 + 77.0i)T^{2} \)
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

−11.53669328350858688176989978817, −11.12476806793919024887274025901, −9.999713782578083179177415430791, −9.181433715806248876277026246620, −7.937809060689194333426282631233, −6.57934841499946031168061867119, −5.18746264403366801942128825350, −4.35420026792377673070919680981, −3.23842223531826032940848586930, −1.63366828595662991941589548408, 1.80962384155512714388489781537, 3.85740204545013001073642925246, 5.19541254785176194030133219069, 6.11064985672256352693027405920, 6.82351783077576913782327627009, 7.76171524697917873060628262071, 8.783976315784198986782026817104, 10.34274227187046856232574713029, 10.95308484515876698129623255331, 12.17381318460697289210433987898

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