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Properties

Label 2-90-45.23-c1-0-4
Degree $2$
Conductor $90$
Sign $0.415 + 0.909i$
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 + (1.10 − 1.33i)3-s + (−0.866 − 0.499i)4-s + (0.792 + 2.09i)5-s + (−1 − 1.41i)6-s + (−1.05 − 0.283i)7-s + (−0.707 + 0.707i)8-s + (−0.548 − 2.94i)9-s + (2.22 − 0.224i)10-s + (−5.44 + 3.14i)11-s + (−1.62 + 0.599i)12-s + (3.34 − 0.896i)13-s + (−0.548 + 0.949i)14-s + (3.66 + 1.25i)15-s + (0.500 + 0.866i)16-s + (3.14 + 3.14i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.639 − 0.769i)3-s + (−0.433 − 0.249i)4-s + (0.354 + 0.935i)5-s + (−0.408 − 0.577i)6-s + (−0.400 − 0.107i)7-s + (−0.249 + 0.249i)8-s + (−0.182 − 0.983i)9-s + (0.703 − 0.0710i)10-s + (−1.64 + 0.948i)11-s + (−0.469 + 0.173i)12-s + (0.928 − 0.248i)13-s + (−0.146 + 0.253i)14-s + (0.945 + 0.325i)15-s + (0.125 + 0.216i)16-s + (0.763 + 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $0.415 + 0.909i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{90} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 90,\ (\ :1/2),\ 0.415 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00054 - 0.642584i\)
\(L(\frac12)\) \(\approx\) \(1.00054 - 0.642584i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.10 + 1.33i)T \)
5 \( 1 + (-0.792 - 2.09i)T \)
good7 \( 1 + (1.05 + 0.283i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (5.44 - 3.14i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.34 + 0.896i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-3.14 - 3.14i)T + 17iT^{2} \)
19 \( 1 + 1.55iT - 19T^{2} \)
23 \( 1 + (-0.258 - 0.965i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.57 + 2.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.22 + 3.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + (3.39 + 1.96i)T + (20.5 + 35.5i)T^{2} \)
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 \( 1 + (-6.61 + 6.61i)T - 53iT^{2} \)
59 \( 1 + (5.90 - 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.72 - 4.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.978 - 3.65i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 0.635iT - 71T^{2} \)
73 \( 1 + (-2.89 - 2.89i)T + 73iT^{2} \)
79 \( 1 + (-2.12 + 1.22i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.531 + 0.142i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 2.36T + 89T^{2} \)
97 \( 1 + (-10.7 - 2.89i)T + (84.0 + 48.5i)T^{2} \)
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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

−13.46168574007783242796283772005, −13.19721514913448812070901583313, −11.95208062028998333497872413440, −10.55665644342464637104919813184, −9.843605478870605902023211581255, −8.276329126546458989986701159468, −7.13434614338454489354613926847, −5.74882234926733652262896832916, −3.48634924429859908949961157423, −2.24765046214370218406016682020, 3.21760879772961501756350908526, 4.89563015544137714844409378790, 5.83392848652815130925357164227, 7.87056907404020607954339827969, 8.664617076031789117474486139844, 9.668001538673071323711019387261, 10.84646817136597830738098800211, 12.61871575824683142840296067131, 13.54036232040626821560946399402, 14.17178645629577791418049062055

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