Properties

Label 2-285-15.2-c1-0-23
Degree $2$
Conductor $285$
Sign $-0.0299 + 0.999i$
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 + (−1.86 − 1.23i)5-s + (−1.73 − 1.73i)6-s + (−0.633 − 0.633i)7-s + (−2 − 2i)8-s − 2.99·9-s + (3.09 − 0.633i)10-s − 3.73i·11-s + (−0.267 + 0.267i)13-s + 1.26·14-s + (2.13 − 3.23i)15-s + 4·16-s + (−5.09 + 5.09i)17-s + (2.99 − 2.99i)18-s i·19-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + 0.999i·3-s + (−0.834 − 0.550i)5-s + (−0.707 − 0.707i)6-s + (−0.239 − 0.239i)7-s + (−0.707 − 0.707i)8-s − 0.999·9-s + (0.979 − 0.200i)10-s − 1.12i·11-s + (−0.0743 + 0.0743i)13-s + 0.338·14-s + (0.550 − 0.834i)15-s + 16-s + (−1.23 + 1.23i)17-s + (0.707 − 0.707i)18-s − 0.229i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0299 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.0299 + 0.999i$
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: \((2,\ 285,\ (\ :1/2),\ -0.0299 + 0.999i)\)

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 \( 1 + (1.86 + 1.23i)T \)
19 \( 1 + iT \)
good2 \( 1 + (1 - i)T - 2iT^{2} \)
7 \( 1 + (0.633 + 0.633i)T + 7iT^{2} \)
11 \( 1 + 3.73iT - 11T^{2} \)
13 \( 1 + (0.267 - 0.267i)T - 13iT^{2} \)
17 \( 1 + (5.09 - 5.09i)T - 17iT^{2} \)
23 \( 1 + (-0.464 - 0.464i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 5.26T + 31T^{2} \)
37 \( 1 + (4.46 + 4.46i)T + 37iT^{2} \)
41 \( 1 + 9.66iT - 41T^{2} \)
43 \( 1 + (5.09 - 5.09i)T - 43iT^{2} \)
47 \( 1 + (8.56 - 8.56i)T - 47iT^{2} \)
53 \( 1 + (-5.73 - 5.73i)T + 53iT^{2} \)
59 \( 1 + 0.196T + 59T^{2} \)
61 \( 1 + 7.73T + 61T^{2} \)
67 \( 1 + (2.19 + 2.19i)T + 67iT^{2} \)
71 \( 1 - 10.7iT - 71T^{2} \)
73 \( 1 + (-5.36 + 5.36i)T - 73iT^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + (0.267 + 0.267i)T + 83iT^{2} \)
89 \( 1 + 7.26T + 89T^{2} \)
97 \( 1 + (6.46 + 6.46i)T + 97iT^{2} \)
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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

−11.32602434182705133029176505456, −10.62111028263695793713691470955, −9.291235223981493460762044158230, −8.706345745187326377625743366356, −8.053689027011803063585337973799, −6.79543816577951223815609206356, −5.63616021400013802762425419974, −4.18354881997924089090773445263, −3.38133749014681631833085232182, 0, 1.96682305574625496043548189915, 3.05669547926487594506152078724, 4.96937156796255650927566361660, 6.50189260558032484391225240322, 7.24753696109469498766867985766, 8.322126659000718580704279262046, 9.243531033504562406818803210586, 10.27208178766484561187720266136, 11.32996779399947294963906966375

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