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Properties

Label 2-90-45.32-c1-0-2
Degree $2$
Conductor $90$
Sign $0.910 + 0.413i$
Analytic cond. $0.718653$
Root an. cond. $0.847734$
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  + (−0.965 − 0.258i)2-s + (1.73 − 0.0795i)3-s + (0.866 + 0.499i)4-s + (−0.139 − 2.23i)5-s + (−1.69 − 0.370i)6-s + (−0.622 + 2.32i)7-s + (−0.707 − 0.707i)8-s + (2.98 − 0.275i)9-s + (−0.442 + 2.19i)10-s + (0.991 − 0.572i)11-s + (1.53 + 0.796i)12-s + (−0.640 − 2.38i)13-s + (1.20 − 2.08i)14-s + (−0.419 − 3.85i)15-s + (0.500 + 0.866i)16-s + (−4.99 + 4.99i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.998 − 0.0459i)3-s + (0.433 + 0.249i)4-s + (−0.0625 − 0.998i)5-s + (−0.690 − 0.151i)6-s + (−0.235 + 0.877i)7-s + (−0.249 − 0.249i)8-s + (0.995 − 0.0917i)9-s + (−0.139 + 0.693i)10-s + (0.299 − 0.172i)11-s + (0.444 + 0.229i)12-s + (−0.177 − 0.662i)13-s + (0.321 − 0.556i)14-s + (−0.108 − 0.994i)15-s + (0.125 + 0.216i)16-s + (−1.21 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.910 + 0.413i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :1/2),\ 0.910 + 0.413i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.925361 - 0.200331i\)
\(L(\frac12)\) \(\approx\) \(0.925361 - 0.200331i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (-1.73 + 0.0795i)T \)
5 \( 1 + (0.139 + 2.23i)T \)
good7 \( 1 + (0.622 - 2.32i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.991 + 0.572i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.640 + 2.38i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.99 - 4.99i)T - 17iT^{2} \)
19 \( 1 - 2.78iT - 19T^{2} \)
23 \( 1 + (5.95 - 1.59i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.672 - 1.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.25 + 2.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.16 + 8.16i)T + 37iT^{2} \)
41 \( 1 + (1.70 + 0.986i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.68 - 2.32i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-11.9 - 3.19i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.84 - 1.84i)T + 53iT^{2} \)
59 \( 1 + (-1.31 + 2.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.54 + 6.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0545 + 0.0146i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.10iT - 71T^{2} \)
73 \( 1 + (-7.82 + 7.82i)T - 73iT^{2} \)
79 \( 1 + (8.46 - 4.88i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.724 + 2.70i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 4.87T + 89T^{2} \)
97 \( 1 + (-2.08 + 7.79i)T + (-84.0 - 48.5i)T^{2} \)
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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

−14.01073134925667920696236611507, −12.74306332557544938543694238433, −12.21421308285627499876696844307, −10.52358281527727295567243035437, −9.286081019599070198760433679882, −8.665835085709969227483474164679, −7.75261271176843941601397077344, −5.95432703888315906091299082564, −3.94385227991878896810115344545, −2.05960826933316984038458829157, 2.48591233655654020821565333034, 4.15354124676156284223049845686, 6.79166298857249226970535629295, 7.25023428102303739735652102875, 8.694231410214484027513786808160, 9.770874476507330238769870890637, 10.60541547789062524520146388529, 11.85276309359446726548946683062, 13.69117687448778746084985936796, 14.06587537148182765242870963226

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