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Properties

Label 2-90-45.2-c1-0-1
Degree $2$
Conductor $90$
Sign $0.947 - 0.319i$
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  + (−0.258 − 0.965i)2-s + (−0.0795 + 1.73i)3-s + (−0.866 + 0.499i)4-s + (0.661 + 2.13i)5-s + (1.69 − 0.370i)6-s + (3.75 − 1.00i)7-s + (0.707 + 0.707i)8-s + (−2.98 − 0.275i)9-s + (1.89 − 1.19i)10-s + (−3.44 − 1.98i)11-s + (−0.796 − 1.53i)12-s + (0.956 + 0.256i)13-s + (−1.94 − 3.36i)14-s + (−3.74 + 0.974i)15-s + (0.500 − 0.866i)16-s + (−0.120 + 0.120i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.0459 + 0.998i)3-s + (−0.433 + 0.249i)4-s + (0.295 + 0.955i)5-s + (0.690 − 0.151i)6-s + (1.41 − 0.380i)7-s + (0.249 + 0.249i)8-s + (−0.995 − 0.0917i)9-s + (0.598 − 0.376i)10-s + (−1.03 − 0.599i)11-s + (−0.229 − 0.444i)12-s + (0.265 + 0.0710i)13-s + (−0.519 − 0.899i)14-s + (−0.967 + 0.251i)15-s + (0.125 − 0.216i)16-s + (−0.0291 + 0.0291i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.947 - 0.319i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :1/2),\ 0.947 - 0.319i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.920564 + 0.151262i\)
\(L(\frac12)\) \(\approx\) \(0.920564 + 0.151262i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (0.0795 - 1.73i)T \)
5 \( 1 + (-0.661 - 2.13i)T \)
good7 \( 1 + (-3.75 + 1.00i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (3.44 + 1.98i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.956 - 0.256i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.120 - 0.120i)T - 17iT^{2} \)
19 \( 1 + 1.88iT - 19T^{2} \)
23 \( 1 + (-1.36 + 5.08i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.15 - 3.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.70 + 8.14i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.26 - 3.26i)T + 37iT^{2} \)
41 \( 1 + (-7.15 + 4.13i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.533 - 1.99i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.897 - 3.34i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.66 + 3.66i)T + 53iT^{2} \)
59 \( 1 + (-2.72 - 4.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.35 - 7.54i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.10 - 7.86i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 6.94iT - 71T^{2} \)
73 \( 1 + (8.27 - 8.27i)T - 73iT^{2} \)
79 \( 1 + (11.7 + 6.78i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.75 + 1.81i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 4.87T + 89T^{2} \)
97 \( 1 + (-1.44 + 0.387i)T + (84.0 - 48.5i)T^{2} \)
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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

−14.32963689902960724985949060259, −13.23648543805560862225761915740, −11.35702795268597906285999251280, −10.98384114021111656623357494192, −10.19388475575991369234523285945, −8.825428167125322110728291815423, −7.67291254090949274413504052014, −5.62503445204123225441378922322, −4.28407584483394434470350031662, −2.68100260688780801836488358418, 1.73428787292598141776655761801, 4.94451411667770672973877522933, 5.76611260963748113229591339883, 7.54672279069282917293181562776, 8.159730414949622211855171825428, 9.241691220189246874914387106744, 10.95588842666517222264581336446, 12.19657517282070384268762320802, 13.08588362641917830736271997575, 14.02715428475129943885560188053

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