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Properties

Label 2-285-19.7-c1-0-5
Degree $2$
Conductor $285$
Sign $0.910 - 0.412i$
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  + (0.207 + 0.358i)2-s + (0.5 + 0.866i)3-s + (0.914 − 1.58i)4-s + (0.5 + 0.866i)5-s + (−0.207 + 0.358i)6-s + 1.82·7-s + 1.58·8-s + (−0.499 + 0.866i)9-s + (−0.207 + 0.358i)10-s − 2.82·11-s + 1.82·12-s + (0.914 − 1.58i)13-s + (0.378 + 0.655i)14-s + (−0.499 + 0.866i)15-s + (−1.49 − 2.59i)16-s + (0.585 + 1.01i)17-s + ⋯
L(s)  = 1  + (0.146 + 0.253i)2-s + (0.288 + 0.499i)3-s + (0.457 − 0.791i)4-s + (0.223 + 0.387i)5-s + (−0.0845 + 0.146i)6-s + 0.691·7-s + 0.560·8-s + (−0.166 + 0.288i)9-s + (−0.0654 + 0.113i)10-s − 0.852·11-s + 0.527·12-s + (0.253 − 0.439i)13-s + (0.101 + 0.175i)14-s + (−0.129 + 0.223i)15-s + (−0.374 − 0.649i)16-s + (0.142 + 0.246i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.910 - 0.412i$
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: \((2,\ 285,\ (\ :1/2),\ 0.910 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73675 + 0.375360i\)
\(L(\frac12)\) \(\approx\) \(1.73675 + 0.375360i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4 - 1.73i)T \)
good2 \( 1 + (-0.207 - 0.358i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 - 1.82T + 7T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 + (-0.914 + 1.58i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.585 - 1.01i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.414 + 0.717i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.82 - 8.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 + 2.17T + 37T^{2} \)
41 \( 1 + (1.41 + 2.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.91 + 6.77i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.58 + 2.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.15 + 7.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.74 - 4.75i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5 + 8.66i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.74 + 8.21i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.67 - 2.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (6.24 - 10.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3 - 5.19i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.62075009669872249607781971805, −10.68716828239377558453516247183, −10.30189365354054438780347394019, −9.136217057903189396273248342487, −7.918550472571814294428617188433, −7.03783851096672845128185869822, −5.61132187096502854683716130706, −5.08596275089432113087427880404, −3.38309768630268570955118127202, −1.85927664038646741948367273824, 1.77082502184961863227353430648, 3.00517892019937582283499914898, 4.43315563257410613202038051105, 5.71716655994584507347787726499, 7.16736792364550433085930013339, 7.83511083891169464320067762999, 8.697558677410357641512976589855, 9.869046827766105125418866877881, 11.26590681707642500500537384126, 11.65231603703479225286962218093

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