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Properties

Label 2-90-45.32-c1-0-5
Degree $2$
Conductor $90$
Sign $0.970 + 0.241i$
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 + (−2.22 − 0.210i)5-s + (1.27 − 1.16i)6-s + (−0.521 + 1.94i)7-s + (0.707 + 0.707i)8-s + (−1.25 − 2.72i)9-s + (−2.09 − 0.779i)10-s + (−1.70 + 0.984i)11-s + (1.53 − 0.796i)12-s + (1.05 + 3.92i)13-s + (−1.00 + 1.74i)14-s + (−2.38 + 3.05i)15-s + (0.500 + 0.866i)16-s + (2.35 − 2.35i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.539 − 0.842i)3-s + (0.433 + 0.249i)4-s + (−0.995 − 0.0942i)5-s + (0.522 − 0.476i)6-s + (−0.197 + 0.736i)7-s + (0.249 + 0.249i)8-s + (−0.418 − 0.908i)9-s + (−0.662 − 0.246i)10-s + (−0.514 + 0.296i)11-s + (0.444 − 0.229i)12-s + (0.291 + 1.08i)13-s + (−0.269 + 0.466i)14-s + (−0.616 + 0.787i)15-s + (0.125 + 0.216i)16-s + (0.572 − 0.572i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.970 + 0.241i$
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: \((2,\ 90,\ (\ :1/2),\ 0.970 + 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36376 - 0.166895i\)
\(L(\frac12)\) \(\approx\) \(1.36376 - 0.166895i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 0.210i)T \)
good7 \( 1 + (0.521 - 1.94i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.70 - 0.984i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.05 - 3.92i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-2.35 + 2.35i)T - 17iT^{2} \)
19 \( 1 + 3.70iT - 19T^{2} \)
23 \( 1 + (6.05 - 1.62i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.74 + 6.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.48 + 6.04i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.26 - 4.26i)T + 37iT^{2} \)
41 \( 1 + (-6.13 - 3.54i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.09 + 2.43i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-7.49 - 2.00i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-7.03 - 7.03i)T + 53iT^{2} \)
59 \( 1 + (-1.34 + 2.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.37 + 7.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.18 - 2.19i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.68iT - 71T^{2} \)
73 \( 1 + (-1.14 + 1.14i)T - 73iT^{2} \)
79 \( 1 + (10.0 - 5.80i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.440 - 1.64i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 2.04T + 89T^{2} \)
97 \( 1 + (-2.60 + 9.71i)T + (-84.0 - 48.5i)T^{2} \)
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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

−13.96027856464549760364042502919, −13.09559632320528467789870257467, −11.96834174409633152322071004812, −11.55984880446351189933468688980, −9.430361765285760861297004396465, −8.162315258809296139594649527987, −7.28033951315223620773979660609, −5.99345562269768015513279761027, −4.19993852546629065683394378293, −2.59758594820897388503820025075, 3.24414664008117669448514210312, 4.09583677000796742339092850936, 5.60027414188458140727662036094, 7.52274720129954352861200684514, 8.427541330882447641857029951900, 10.31432511492175483454994881310, 10.66859833482839975679205238925, 12.11088207473277873107977593277, 13.19856269370687314248402722393, 14.31298212003126384897969386363

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