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Properties

Label 2-285-15.2-c1-0-0
Degree $2$
Conductor $285$
Sign $-0.521 - 0.853i$
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  + (−0.0307 + 0.0307i)2-s + (−1.45 − 0.939i)3-s + 1.99i·4-s + (−1.20 − 1.88i)5-s + (0.0737 − 0.0158i)6-s + (−0.715 − 0.715i)7-s + (−0.123 − 0.123i)8-s + (1.23 + 2.73i)9-s + (0.0950 + 0.0209i)10-s + 5.75i·11-s + (1.87 − 2.90i)12-s + (−1.65 + 1.65i)13-s + 0.0440·14-s + (−0.0204 + 3.87i)15-s − 3.98·16-s + (−4.22 + 4.22i)17-s + ⋯
L(s)  = 1  + (−0.0217 + 0.0217i)2-s + (−0.840 − 0.542i)3-s + 0.999i·4-s + (−0.538 − 0.842i)5-s + (0.0301 − 0.00647i)6-s + (−0.270 − 0.270i)7-s + (−0.0435 − 0.0435i)8-s + (0.411 + 0.911i)9-s + (0.0300 + 0.00663i)10-s + 1.73i·11-s + (0.542 − 0.839i)12-s + (−0.457 + 0.457i)13-s + 0.0117·14-s + (−0.00528 + 0.999i)15-s − 0.997·16-s + (−1.02 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.521 - 0.853i$
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: \((2,\ 285,\ (\ :1/2),\ -0.521 - 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.205224 + 0.365815i\)
\(L(\frac12)\) \(\approx\) \(0.205224 + 0.365815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.45 + 0.939i)T \)
5 \( 1 + (1.20 + 1.88i)T \)
19 \( 1 + iT \)
good2 \( 1 + (0.0307 - 0.0307i)T - 2iT^{2} \)
7 \( 1 + (0.715 + 0.715i)T + 7iT^{2} \)
11 \( 1 - 5.75iT - 11T^{2} \)
13 \( 1 + (1.65 - 1.65i)T - 13iT^{2} \)
17 \( 1 + (4.22 - 4.22i)T - 17iT^{2} \)
23 \( 1 + (1.96 + 1.96i)T + 23iT^{2} \)
29 \( 1 + 5.38T + 29T^{2} \)
31 \( 1 - 4.85T + 31T^{2} \)
37 \( 1 + (-6.33 - 6.33i)T + 37iT^{2} \)
41 \( 1 - 1.84iT - 41T^{2} \)
43 \( 1 + (1.35 - 1.35i)T - 43iT^{2} \)
47 \( 1 + (-5.93 + 5.93i)T - 47iT^{2} \)
53 \( 1 + (0.615 + 0.615i)T + 53iT^{2} \)
59 \( 1 + 2.01T + 59T^{2} \)
61 \( 1 + 5.22T + 61T^{2} \)
67 \( 1 + (8.61 + 8.61i)T + 67iT^{2} \)
71 \( 1 - 8.55iT - 71T^{2} \)
73 \( 1 + (8.09 - 8.09i)T - 73iT^{2} \)
79 \( 1 + 13.6iT - 79T^{2} \)
83 \( 1 + (1.52 + 1.52i)T + 83iT^{2} \)
89 \( 1 - 7.80T + 89T^{2} \)
97 \( 1 + (-9.06 - 9.06i)T + 97iT^{2} \)
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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

−12.06517882665702346898514426245, −11.68632375198309555442310608862, −10.36696202676568984809782187734, −9.211352359195259589440545815092, −8.064030125223348516685359351147, −7.30971436843995634904848202543, −6.47791600533540407508295164440, −4.73823347318525118944755595394, −4.19663426270414439430596156735, −2.04024993259307952230961699061, 0.33795905928910970674394909419, 2.92389628445323892242128973846, 4.34161765353873609923988575899, 5.69660743562656192516489001836, 6.20066998955987785404487305302, 7.38995234848022977493397444661, 8.949358974004826903733131133684, 9.809958794935488805292402198592, 10.87919938748248108143532044497, 11.14019508627340108715791754999

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