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Properties

Label 2-90-45.2-c1-0-4
Degree $2$
Conductor $90$
Sign $0.615 + 0.787i$
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 + (1.62 − 0.599i)3-s + (−0.866 + 0.499i)4-s + (2.20 + 0.358i)5-s + (−1 − 1.41i)6-s + (−4.40 + 1.18i)7-s + (0.707 + 0.707i)8-s + (2.28 − 1.94i)9-s + (−0.224 − 2.22i)10-s + (−0.550 − 0.317i)11-s + (−1.10 + 1.33i)12-s + (−3.34 − 0.896i)13-s + (2.28 + 3.94i)14-s + (3.80 − 0.741i)15-s + (0.500 − 0.866i)16-s + (0.317 − 0.317i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.938 − 0.346i)3-s + (−0.433 + 0.249i)4-s + (0.987 + 0.160i)5-s + (−0.408 − 0.577i)6-s + (−1.66 + 0.446i)7-s + (0.249 + 0.249i)8-s + (0.760 − 0.649i)9-s + (−0.0710 − 0.703i)10-s + (−0.165 − 0.0958i)11-s + (−0.319 + 0.384i)12-s + (−0.928 − 0.248i)13-s + (0.609 + 1.05i)14-s + (0.981 − 0.191i)15-s + (0.125 − 0.216i)16-s + (0.0770 − 0.0770i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.615 + 0.787i$
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: \((2,\ 90,\ (\ :1/2),\ 0.615 + 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00182 - 0.488464i\)
\(L(\frac12)\) \(\approx\) \(1.00182 - 0.488464i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 0.599i)T \)
5 \( 1 + (-2.20 - 0.358i)T \)
good7 \( 1 + (4.40 - 1.18i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.550 + 0.317i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.34 + 0.896i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.317 + 0.317i)T - 17iT^{2} \)
19 \( 1 - 6.44iT - 19T^{2} \)
23 \( 1 + (0.258 - 0.965i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.158 - 0.275i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.224 + 0.389i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 + 3i)T + 37iT^{2} \)
41 \( 1 + (-6.39 + 3.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.896 + 3.34i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.32 + 8.69i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.78 - 3.78i)T + 53iT^{2} \)
59 \( 1 + (4.48 + 7.77i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.275 + 0.476i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.71 - 6.38i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 6.29iT - 71T^{2} \)
73 \( 1 + (6.89 - 6.89i)T - 73iT^{2} \)
79 \( 1 + (2.12 + 1.22i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.26 + 1.41i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 8.02T + 89T^{2} \)
97 \( 1 + (2.59 - 0.695i)T + (84.0 - 48.5i)T^{2} \)
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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

−13.75390838752650612222274233985, −12.82975141676855669380480755959, −12.28544385574707569462784434033, −10.18899968214190585881307450664, −9.738465556796882741992324694516, −8.761182982633527232832837116383, −7.19338605129095804100231117999, −5.82559861955877040318123449483, −3.45443420619949451364967755639, −2.31256053256784956475585516413, 2.85018954803007261721190497747, 4.69661118540076006162861913531, 6.37784630758992714555097048811, 7.37512029690529663562849923539, 9.032593130855053680423076343793, 9.608040547364422430860932352312, 10.40793916549395773361012592721, 12.77800970167359361823309551339, 13.37567498932463299362309710398, 14.25340444945879559499823752920

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