Properties

Label 2-285-15.2-c1-0-26
Degree $2$
Conductor $285$
Sign $-0.880 + 0.473i$
Analytic cond. $2.27573$
Root an. cond. $1.50855$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s − 1.73i·3-s + (−0.133 + 2.23i)5-s + (1.73 + 1.73i)6-s + (−2.36 − 2.36i)7-s + (−2 − 2i)8-s − 2.99·9-s + (−2.09 − 2.36i)10-s − 0.267i·11-s + (−3.73 + 3.73i)13-s + 4.73·14-s + (3.86 + 0.232i)15-s + 4·16-s + (0.0980 − 0.0980i)17-s + (2.99 − 2.99i)18-s i·19-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 0.999i·3-s + (−0.0599 + 0.998i)5-s + (0.707 + 0.707i)6-s + (−0.894 − 0.894i)7-s + (−0.707 − 0.707i)8-s − 0.999·9-s + (−0.663 − 0.748i)10-s − 0.0807i·11-s + (−1.03 + 1.03i)13-s + 1.26·14-s + (0.998 + 0.0599i)15-s + 16-s + (0.0237 − 0.0237i)17-s + (0.707 − 0.707i)18-s − 0.229i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.880 + 0.473i$
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: \(1\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ -0.880 + 0.473i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
5 \( 1 + (0.133 - 2.23i)T \)
19 \( 1 + iT \)
good2 \( 1 + (1 - i)T - 2iT^{2} \)
7 \( 1 + (2.36 + 2.36i)T + 7iT^{2} \)
11 \( 1 + 0.267iT - 11T^{2} \)
13 \( 1 + (3.73 - 3.73i)T - 13iT^{2} \)
17 \( 1 + (-0.0980 + 0.0980i)T - 17iT^{2} \)
23 \( 1 + (6.46 + 6.46i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8.73T + 31T^{2} \)
37 \( 1 + (-2.46 - 2.46i)T + 37iT^{2} \)
41 \( 1 - 7.66iT - 41T^{2} \)
43 \( 1 + (-0.0980 + 0.0980i)T - 43iT^{2} \)
47 \( 1 + (-3.56 + 3.56i)T - 47iT^{2} \)
53 \( 1 + (-2.26 - 2.26i)T + 53iT^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 + 4.26T + 61T^{2} \)
67 \( 1 + (-8.19 - 8.19i)T + 67iT^{2} \)
71 \( 1 - 7.26iT - 71T^{2} \)
73 \( 1 + (-3.63 + 3.63i)T - 73iT^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 + (3.73 + 3.73i)T + 83iT^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + (-0.464 - 0.464i)T + 97iT^{2} \)
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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

−11.51658558599359597930947292619, −10.27064526215938628534665232718, −9.448513247399149978996680005966, −8.248023996354336233275263751365, −7.24952004638161572618194194344, −6.88836176373188395397098630786, −6.11027505599918245012697588266, −3.87048728869692670554750500763, −2.54629852999416177505490431095, 0, 2.32472584880068438846390550570, 3.67963559423411933433291356385, 5.37945806336755436660238095537, 5.70830687301431642374585502297, 7.893441505969545622899723632756, 8.899213621506554020088788970383, 9.581944189835479392546629492363, 9.991993754636326194737573145842, 11.13281312230291030926851314027

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