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Properties

Label 2-90-45.32-c1-0-1
Degree $2$
Conductor $90$
Sign $0.512 - 0.858i$
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 + (−0.933 + 1.45i)3-s + (0.866 + 0.499i)4-s + (−0.847 + 2.06i)5-s + (−1.27 + 1.16i)6-s + (0.686 − 2.56i)7-s + (0.707 + 0.707i)8-s + (−1.25 − 2.72i)9-s + (−1.35 + 1.77i)10-s + (4.15 − 2.39i)11-s + (−1.53 + 0.796i)12-s + (−0.155 − 0.581i)13-s + (1.32 − 2.29i)14-s + (−2.22 − 3.16i)15-s + (0.500 + 0.866i)16-s + (−4.40 + 4.40i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.539 + 0.842i)3-s + (0.433 + 0.249i)4-s + (−0.378 + 0.925i)5-s + (−0.522 + 0.476i)6-s + (0.259 − 0.968i)7-s + (0.249 + 0.249i)8-s + (−0.418 − 0.908i)9-s + (−0.428 + 0.562i)10-s + (1.25 − 0.723i)11-s + (−0.444 + 0.229i)12-s + (−0.0432 − 0.161i)13-s + (0.354 − 0.613i)14-s + (−0.575 − 0.818i)15-s + (0.125 + 0.216i)16-s + (−1.06 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.512 - 0.858i$
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: \((2,\ 90,\ (\ :1/2),\ 0.512 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02598 + 0.582768i\)
\(L(\frac12)\) \(\approx\) \(1.02598 + 0.582768i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.933 - 1.45i)T \)
5 \( 1 + (0.847 - 2.06i)T \)
good7 \( 1 + (-0.686 + 2.56i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-4.15 + 2.39i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.155 + 0.581i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.40 - 4.40i)T - 17iT^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 + (2.54 - 0.681i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.920 - 1.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.03 - 3.53i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.632 - 0.632i)T + 37iT^{2} \)
41 \( 1 + (5.58 + 3.22i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.40 - 0.644i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (3.82 + 1.02i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.31 - 1.31i)T + 53iT^{2} \)
59 \( 1 + (-0.0645 + 0.111i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.27 - 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.6 - 2.85i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (-3.30 + 3.30i)T - 73iT^{2} \)
79 \( 1 + (3.62 - 2.09i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.97 + 11.1i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 2.04T + 89T^{2} \)
97 \( 1 + (-4.47 + 16.7i)T + (-84.0 - 48.5i)T^{2} \)
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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

−14.44904642136309102131070382053, −13.50079714365832351165480199581, −11.89024631487305082354442798108, −11.07751986454699943615663180555, −10.43068302144441036705480940785, −8.765278131477638518765945984077, −7.04074545161929798602164166554, −6.17143034314061884723906545851, −4.41876549374962844146826455545, −3.52159760970106920677294803102, 1.90223175040572273816067183865, 4.39704058301326909823845242000, 5.59171819035744028787689697723, 6.81535600359912997296376651946, 8.222828315608335762795548526679, 9.489515125274501931606204327757, 11.42603437347742652147869006118, 11.96546516326445339180375226954, 12.62376861613516320773903494244, 13.74034609505766088553116401491

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