Properties

Label 2-285-15.2-c1-0-18
Degree $2$
Conductor $285$
Sign $0.668 - 0.743i$
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  + (−1.53 + 1.53i)2-s + (1.34 − 1.09i)3-s − 2.69i·4-s + (2.05 + 0.869i)5-s + (−0.376 + 3.73i)6-s + (1.55 + 1.55i)7-s + (1.06 + 1.06i)8-s + (0.598 − 2.93i)9-s + (−4.49 + 1.82i)10-s − 2.52i·11-s + (−2.95 − 3.61i)12-s + (−0.427 + 0.427i)13-s − 4.75·14-s + (3.71 − 1.09i)15-s + 2.11·16-s + (1.45 − 1.45i)17-s + ⋯
L(s)  = 1  + (−1.08 + 1.08i)2-s + (0.774 − 0.632i)3-s − 1.34i·4-s + (0.921 + 0.389i)5-s + (−0.153 + 1.52i)6-s + (0.586 + 0.586i)7-s + (0.377 + 0.377i)8-s + (0.199 − 0.979i)9-s + (−1.41 + 0.576i)10-s − 0.760i·11-s + (−0.853 − 1.04i)12-s + (−0.118 + 0.118i)13-s − 1.27·14-s + (0.959 − 0.281i)15-s + 0.529·16-s + (0.353 − 0.353i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.668 - 0.743i$
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: \((2,\ 285,\ (\ :1/2),\ 0.668 - 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04693 + 0.466830i\)
\(L(\frac12)\) \(\approx\) \(1.04693 + 0.466830i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.34 + 1.09i)T \)
5 \( 1 + (-2.05 - 0.869i)T \)
19 \( 1 - iT \)
good2 \( 1 + (1.53 - 1.53i)T - 2iT^{2} \)
7 \( 1 + (-1.55 - 1.55i)T + 7iT^{2} \)
11 \( 1 + 2.52iT - 11T^{2} \)
13 \( 1 + (0.427 - 0.427i)T - 13iT^{2} \)
17 \( 1 + (-1.45 + 1.45i)T - 17iT^{2} \)
23 \( 1 + (-3.56 - 3.56i)T + 23iT^{2} \)
29 \( 1 + 6.37T + 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 + (-6.17 - 6.17i)T + 37iT^{2} \)
41 \( 1 + 4.29iT - 41T^{2} \)
43 \( 1 + (2.42 - 2.42i)T - 43iT^{2} \)
47 \( 1 + (2.49 - 2.49i)T - 47iT^{2} \)
53 \( 1 + (-8.40 - 8.40i)T + 53iT^{2} \)
59 \( 1 + 0.181T + 59T^{2} \)
61 \( 1 + 14.9T + 61T^{2} \)
67 \( 1 + (6.58 + 6.58i)T + 67iT^{2} \)
71 \( 1 + 14.8iT - 71T^{2} \)
73 \( 1 + (-4.76 + 4.76i)T - 73iT^{2} \)
79 \( 1 - 7.82iT - 79T^{2} \)
83 \( 1 + (10.7 + 10.7i)T + 83iT^{2} \)
89 \( 1 - 4.99T + 89T^{2} \)
97 \( 1 + (4.73 + 4.73i)T + 97iT^{2} \)
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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

−11.92733490223178208714340860258, −10.70238183301792248006387657774, −9.406275351214678272054480557640, −9.112073321780941301628086535613, −8.038576819794990111581622394872, −7.31909553538415307247984430158, −6.28306659326306577936264676176, −5.48582849377255511211142894648, −3.10007945594175048115740383568, −1.54033223615391468569101297673, 1.55818617186886523952515705903, 2.62072829093817816998542402188, 4.13620455176154853696624538482, 5.37220017125008528823622301712, 7.32854896063191581059404868930, 8.318479957785742412549290916442, 9.160904737064618277672190506577, 9.806681455831554175304221518802, 10.51982866637154127547847554651, 11.22673261214108813130714344794

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