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Properties

Label 2-90-5.2-c2-0-3
Degree $2$
Conductor $90$
Sign $0.767 - 0.640i$
Analytic cond. $2.45232$
Root an. cond. $1.56598$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + 2i·4-s + (3 − 4i)5-s + (8 + 8i)7-s + (−2 + 2i)8-s + (7 − i)10-s − 4·11-s + (−3 + 3i)13-s + 16i·14-s − 4·16-s + (−19 − 19i)17-s − 8i·19-s + (8 + 6i)20-s + (−4 − 4i)22-s + (20 − 20i)23-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + (0.600 − 0.800i)5-s + (1.14 + 1.14i)7-s + (−0.250 + 0.250i)8-s + (0.700 − 0.100i)10-s − 0.363·11-s + (−0.230 + 0.230i)13-s + 1.14i·14-s − 0.250·16-s + (−1.11 − 1.11i)17-s − 0.421i·19-s + (0.400 + 0.300i)20-s + (−0.181 − 0.181i)22-s + (0.869 − 0.869i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.767 - 0.640i$
Analytic conductor: \(2.45232\)
Root analytic conductor: \(1.56598\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :1),\ 0.767 - 0.640i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.73252 + 0.627978i\)
\(L(\frac12)\) \(\approx\) \(1.73252 + 0.627978i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
5 \( 1 + (-3 + 4i)T \)
good7 \( 1 + (-8 - 8i)T + 49iT^{2} \)
11 \( 1 + 4T + 121T^{2} \)
13 \( 1 + (3 - 3i)T - 169iT^{2} \)
17 \( 1 + (19 + 19i)T + 289iT^{2} \)
19 \( 1 + 8iT - 361T^{2} \)
23 \( 1 + (-20 + 20i)T - 529iT^{2} \)
29 \( 1 - 38iT - 841T^{2} \)
31 \( 1 + 44T + 961T^{2} \)
37 \( 1 + (3 + 3i)T + 1.36e3iT^{2} \)
41 \( 1 + 70T + 1.68e3T^{2} \)
43 \( 1 + (-36 + 36i)T - 1.84e3iT^{2} \)
47 \( 1 + 2.20e3iT^{2} \)
53 \( 1 + (17 - 17i)T - 2.80e3iT^{2} \)
59 \( 1 - 92iT - 3.48e3T^{2} \)
61 \( 1 - 72T + 3.72e3T^{2} \)
67 \( 1 + (-44 - 44i)T + 4.48e3iT^{2} \)
71 \( 1 - 88T + 5.04e3T^{2} \)
73 \( 1 + (-55 + 55i)T - 5.32e3iT^{2} \)
79 \( 1 - 12iT - 6.24e3T^{2} \)
83 \( 1 + (-24 + 24i)T - 6.88e3iT^{2} \)
89 \( 1 + 26iT - 7.92e3T^{2} \)
97 \( 1 + (57 + 57i)T + 9.40e3iT^{2} \)
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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

−14.04069499655929341045975085706, −12.99268275993724894539496173295, −12.08521945752718196173176159383, −11.00036747124215757976324621525, −9.112987347504254366248898166486, −8.569935515762549845422525506963, −7.00050538449587226768249018330, −5.38090472814699909208613571862, −4.81756719324260560081254253982, −2.29470844174315848453011874901, 1.90007058324373184107056649676, 3.79411676590324652133929518839, 5.23742088478129338331876892094, 6.73231911598271029206748655992, 7.986747517728860552229865650803, 9.752002988786092542843397553355, 10.81907659011399430574968761461, 11.23881236266706286156187677290, 12.94633890994636546402095128105, 13.71462172595389698386510033502

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