Properties

Label 2-285-15.2-c1-0-2
Degree $2$
Conductor $285$
Sign $-0.890 - 0.454i$
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  + (−0.637 + 0.637i)2-s + (−1.73 − 0.0191i)3-s + 1.18i·4-s + (1.18 + 1.89i)5-s + (1.11 − 1.09i)6-s + (2.03 + 2.03i)7-s + (−2.03 − 2.03i)8-s + (2.99 + 0.0661i)9-s + (−1.96 − 0.458i)10-s − 2.29i·11-s + (0.0226 − 2.05i)12-s + (−2.17 + 2.17i)13-s − 2.58·14-s + (−2.00 − 3.31i)15-s + 0.219·16-s + (−4.12 + 4.12i)17-s + ⋯
L(s)  = 1  + (−0.450 + 0.450i)2-s + (−0.999 − 0.0110i)3-s + 0.593i·4-s + (0.527 + 0.849i)5-s + (0.455 − 0.445i)6-s + (0.767 + 0.767i)7-s + (−0.718 − 0.718i)8-s + (0.999 + 0.0220i)9-s + (−0.621 − 0.144i)10-s − 0.691i·11-s + (0.00654 − 0.593i)12-s + (−0.603 + 0.603i)13-s − 0.692·14-s + (−0.518 − 0.855i)15-s + 0.0548·16-s + (−1.00 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.890 - 0.454i$
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: \((2,\ 285,\ (\ :1/2),\ -0.890 - 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.163502 + 0.679849i\)
\(L(\frac12)\) \(\approx\) \(0.163502 + 0.679849i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0191i)T \)
5 \( 1 + (-1.18 - 1.89i)T \)
19 \( 1 - iT \)
good2 \( 1 + (0.637 - 0.637i)T - 2iT^{2} \)
7 \( 1 + (-2.03 - 2.03i)T + 7iT^{2} \)
11 \( 1 + 2.29iT - 11T^{2} \)
13 \( 1 + (2.17 - 2.17i)T - 13iT^{2} \)
17 \( 1 + (4.12 - 4.12i)T - 17iT^{2} \)
23 \( 1 + (0.124 + 0.124i)T + 23iT^{2} \)
29 \( 1 + 3.99T + 29T^{2} \)
31 \( 1 - 8.55T + 31T^{2} \)
37 \( 1 + (7.98 + 7.98i)T + 37iT^{2} \)
41 \( 1 - 4.02iT - 41T^{2} \)
43 \( 1 + (-5.82 + 5.82i)T - 43iT^{2} \)
47 \( 1 + (2.25 - 2.25i)T - 47iT^{2} \)
53 \( 1 + (-6.79 - 6.79i)T + 53iT^{2} \)
59 \( 1 - 0.588T + 59T^{2} \)
61 \( 1 - 4.76T + 61T^{2} \)
67 \( 1 + (-3.10 - 3.10i)T + 67iT^{2} \)
71 \( 1 - 4.72iT - 71T^{2} \)
73 \( 1 + (2.06 - 2.06i)T - 73iT^{2} \)
79 \( 1 - 11.8iT - 79T^{2} \)
83 \( 1 + (-10.2 - 10.2i)T + 83iT^{2} \)
89 \( 1 + 0.931T + 89T^{2} \)
97 \( 1 + (-12.3 - 12.3i)T + 97iT^{2} \)
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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

−12.01937846973774133001231158956, −11.31999638200903083062579328563, −10.45944703534542789416724231970, −9.308140903747285918955525487336, −8.358098301624071211123196046091, −7.20665041885011170350236657916, −6.40169043006669104903569053749, −5.52153399359605226728613951536, −4.02173302227332875239027876048, −2.23798578703094845334992758810, 0.68911559228624739084682564483, 1.99596694866841611706307456425, 4.71878538740561073650575565644, 5.03367380540116860516334397486, 6.34952828728729395042943152258, 7.50075408755226171878477884955, 8.851462103926490621310678459140, 9.892993019088147548238630533296, 10.35645760444584120377281648794, 11.39209924850021495039333747121

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