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Properties

Label 2-90-45.34-c1-0-4
Degree $2$
Conductor $90$
Sign $0.964 - 0.262i$
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  + (0.866 + 0.5i)2-s + (1.57 − 0.724i)3-s + (0.499 + 0.866i)4-s + (−1.81 + 1.30i)5-s + (1.72 + 0.158i)6-s + (−3.85 − 2.22i)7-s + 0.999i·8-s + (1.94 − 2.28i)9-s + (−2.22 + 0.224i)10-s + (−0.724 + 1.25i)11-s + (1.41 + i)12-s + (2.12 − 1.22i)13-s + (−2.22 − 3.85i)14-s + (−1.90 + 3.37i)15-s + (−0.5 + 0.866i)16-s + 3.89i·17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.908 − 0.418i)3-s + (0.249 + 0.433i)4-s + (−0.811 + 0.584i)5-s + (0.704 + 0.0648i)6-s + (−1.45 − 0.840i)7-s + 0.353i·8-s + (0.649 − 0.760i)9-s + (−0.703 + 0.0710i)10-s + (−0.218 + 0.378i)11-s + (0.408 + 0.288i)12-s + (0.588 − 0.339i)13-s + (−0.594 − 1.02i)14-s + (−0.492 + 0.870i)15-s + (−0.125 + 0.216i)16-s + 0.945i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.964 - 0.262i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :1/2),\ 0.964 - 0.262i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38736 + 0.185602i\)
\(L(\frac12)\) \(\approx\) \(1.38736 + 0.185602i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-1.57 + 0.724i)T \)
5 \( 1 + (1.81 - 1.30i)T \)
good7 \( 1 + (3.85 + 2.22i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.724 - 1.25i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.12 + 1.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.89iT - 17T^{2} \)
19 \( 1 - 0.550T + 19T^{2} \)
23 \( 1 + (-2.51 + 1.44i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.22 + 5.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.45 - 3.72i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.389 + 0.224i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.44iT - 53T^{2} \)
59 \( 1 + (5.62 + 9.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.224 + 0.389i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.18 + 4.72i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.44T + 71T^{2} \)
73 \( 1 - 4.79iT - 73T^{2} \)
79 \( 1 + (3.67 - 6.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.46 + 2i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + (11.2 + 6.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.17467893097328759480392022460, −13.03498771505224042998651915741, −12.65067253081451093811452447657, −10.98223313798292145897231462838, −9.764053461889174124066589696630, −8.233178006023554614173165979508, −7.19940273509319772460285987318, −6.41039935120279862995092961411, −3.96718856080354666891603288247, −3.13343893345047280814276023466, 2.88985220502493558871282011110, 3.96963944904665922993769664424, 5.55387967066356602848588482080, 7.26795141191650353583965647233, 8.851178632670324600509652836942, 9.461483220451065979342472346512, 10.95230793654349760257052016475, 12.21117089253933826169534465305, 13.03593206412215795745641566182, 13.90259167296091556841429834919

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