Properties

Label 2-285-15.2-c1-0-24
Degree $2$
Conductor $285$
Sign $0.729 + 0.683i$
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  + (0.473 − 0.473i)2-s + (0.589 − 1.62i)3-s + 1.55i·4-s + (1.83 − 1.28i)5-s + (−0.491 − 1.05i)6-s + (1.99 + 1.99i)7-s + (1.68 + 1.68i)8-s + (−2.30 − 1.92i)9-s + (0.258 − 1.47i)10-s − 1.23i·11-s + (2.52 + 0.915i)12-s + (−2.31 + 2.31i)13-s + 1.89·14-s + (−1.01 − 3.73i)15-s − 1.51·16-s + (0.610 − 0.610i)17-s + ⋯
L(s)  = 1  + (0.334 − 0.334i)2-s + (0.340 − 0.940i)3-s + 0.775i·4-s + (0.818 − 0.574i)5-s + (−0.200 − 0.428i)6-s + (0.755 + 0.755i)7-s + (0.594 + 0.594i)8-s + (−0.768 − 0.640i)9-s + (0.0818 − 0.466i)10-s − 0.372i·11-s + (0.729 + 0.264i)12-s + (−0.642 + 0.642i)13-s + 0.505·14-s + (−0.261 − 0.965i)15-s − 0.377·16-s + (0.148 − 0.148i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.729 + 0.683i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ 0.729 + 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78887 - 0.707413i\)
\(L(\frac12)\) \(\approx\) \(1.78887 - 0.707413i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.589 + 1.62i)T \)
5 \( 1 + (-1.83 + 1.28i)T \)
19 \( 1 + iT \)
good2 \( 1 + (-0.473 + 0.473i)T - 2iT^{2} \)
7 \( 1 + (-1.99 - 1.99i)T + 7iT^{2} \)
11 \( 1 + 1.23iT - 11T^{2} \)
13 \( 1 + (2.31 - 2.31i)T - 13iT^{2} \)
17 \( 1 + (-0.610 + 0.610i)T - 17iT^{2} \)
23 \( 1 + (4.45 + 4.45i)T + 23iT^{2} \)
29 \( 1 + 4.01T + 29T^{2} \)
31 \( 1 - 5.53T + 31T^{2} \)
37 \( 1 + (3.36 + 3.36i)T + 37iT^{2} \)
41 \( 1 - 5.42iT - 41T^{2} \)
43 \( 1 + (5.73 - 5.73i)T - 43iT^{2} \)
47 \( 1 + (3.15 - 3.15i)T - 47iT^{2} \)
53 \( 1 + (-7.65 - 7.65i)T + 53iT^{2} \)
59 \( 1 + 5.84T + 59T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 + (1.62 + 1.62i)T + 67iT^{2} \)
71 \( 1 - 4.64iT - 71T^{2} \)
73 \( 1 + (-0.204 + 0.204i)T - 73iT^{2} \)
79 \( 1 - 0.300iT - 79T^{2} \)
83 \( 1 + (-9.55 - 9.55i)T + 83iT^{2} \)
89 \( 1 + 4.28T + 89T^{2} \)
97 \( 1 + (11.9 + 11.9i)T + 97iT^{2} \)
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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

−12.03147287620000401025324890839, −11.22037269243568829018391407279, −9.618688561570858224917791688977, −8.585630332982655199243373320454, −8.106763315478822437624352985994, −6.83535068273644118389379981965, −5.63833905131587786767861694874, −4.48836353515732385547720819443, −2.73086515337929365430997984878, −1.85178086327831338095969954287, 1.98712862295396199906421117477, 3.73488131216230584001274883613, 4.97222816285222530069103099856, 5.64181029000401082695139303838, 6.96164607765873530279370314514, 8.012407585713013299620907116970, 9.464649157360957050497198549625, 10.21580403337885758413408034806, 10.54053106400131840217917414272, 11.69577838013560529716405747503

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