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Properties

Label 2-90-45.38-c1-0-5
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: $\chi_{90} (83, \cdot )$
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

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

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