Properties

Label 2-285-15.2-c1-0-22
Degree $2$
Conductor $285$
Sign $0.973 + 0.226i$
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 + (1.71 − 0.229i)3-s − 2.68i·4-s + (−1.36 − 1.77i)5-s + (−2.27 + 2.98i)6-s + (−1.27 − 1.27i)7-s + (1.05 + 1.05i)8-s + (2.89 − 0.788i)9-s + (4.80 + 0.626i)10-s + 0.550i·11-s + (−0.617 − 4.61i)12-s + (1.35 − 1.35i)13-s + 3.90·14-s + (−2.74 − 2.72i)15-s + 2.14·16-s + (4.95 − 4.95i)17-s + ⋯
L(s)  = 1  + (−1.08 + 1.08i)2-s + (0.991 − 0.132i)3-s − 1.34i·4-s + (−0.609 − 0.792i)5-s + (−0.929 + 1.21i)6-s + (−0.481 − 0.481i)7-s + (0.373 + 0.373i)8-s + (0.964 − 0.262i)9-s + (1.51 + 0.198i)10-s + 0.165i·11-s + (−0.178 − 1.33i)12-s + (0.376 − 0.376i)13-s + 1.04·14-s + (−0.709 − 0.704i)15-s + 0.536·16-s + (1.20 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.973 + 0.226i$
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: \((2,\ 285,\ (\ :1/2),\ 0.973 + 0.226i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.844375 - 0.0969389i\)
\(L(\frac12)\) \(\approx\) \(0.844375 - 0.0969389i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 + 0.229i)T \)
5 \( 1 + (1.36 + 1.77i)T \)
19 \( 1 + iT \)
good2 \( 1 + (1.53 - 1.53i)T - 2iT^{2} \)
7 \( 1 + (1.27 + 1.27i)T + 7iT^{2} \)
11 \( 1 - 0.550iT - 11T^{2} \)
13 \( 1 + (-1.35 + 1.35i)T - 13iT^{2} \)
17 \( 1 + (-4.95 + 4.95i)T - 17iT^{2} \)
23 \( 1 + (6.10 + 6.10i)T + 23iT^{2} \)
29 \( 1 - 3.46T + 29T^{2} \)
31 \( 1 - 4.86T + 31T^{2} \)
37 \( 1 + (1.07 + 1.07i)T + 37iT^{2} \)
41 \( 1 - 9.60iT - 41T^{2} \)
43 \( 1 + (4.90 - 4.90i)T - 43iT^{2} \)
47 \( 1 + (-0.121 + 0.121i)T - 47iT^{2} \)
53 \( 1 + (-0.327 - 0.327i)T + 53iT^{2} \)
59 \( 1 + 5.26T + 59T^{2} \)
61 \( 1 + 2.56T + 61T^{2} \)
67 \( 1 + (0.646 + 0.646i)T + 67iT^{2} \)
71 \( 1 - 0.570iT - 71T^{2} \)
73 \( 1 + (11.1 - 11.1i)T - 73iT^{2} \)
79 \( 1 + 3.73iT - 79T^{2} \)
83 \( 1 + (-6.83 - 6.83i)T + 83iT^{2} \)
89 \( 1 - 14.6T + 89T^{2} \)
97 \( 1 + (0.0917 + 0.0917i)T + 97iT^{2} \)
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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

−11.93395016891784953062493554792, −10.16474129824388477707066834839, −9.673981868425441924343031182724, −8.617965265844612999478419025679, −8.036893627743140477518590621996, −7.31905698451773746017579245628, −6.28263920284072798630705608283, −4.66267263027427024045239359037, −3.25156301677783349908390137110, −0.872569541829313554991371875988, 1.83272966910940458815589039280, 3.14194881456931568142226116573, 3.79378412425316593995039638058, 6.09111497038285715900783195607, 7.59569434283908538911057620143, 8.248871742599495649743717630589, 9.123370570002517155329852270004, 10.12951549630826855194877675614, 10.51196295541588845601688402789, 11.84785355583748998377425123373

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