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Properties

Label 2-90-45.23-c1-0-5
Degree $2$
Conductor $90$
Sign $-0.337 + 0.941i$
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 + (−1.45 − 0.933i)3-s + (−0.866 − 0.499i)4-s + (1.36 − 1.76i)5-s + (−1.27 + 1.16i)6-s + (−2.56 − 0.686i)7-s + (−0.707 + 0.707i)8-s + (1.25 + 2.72i)9-s + (−1.35 − 1.77i)10-s + (4.15 − 2.39i)11-s + (0.796 + 1.53i)12-s + (0.581 − 0.155i)13-s + (−1.32 + 2.29i)14-s + (−3.64 + 1.30i)15-s + (0.500 + 0.866i)16-s + (4.40 + 4.40i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.842 − 0.539i)3-s + (−0.433 − 0.249i)4-s + (0.612 − 0.790i)5-s + (−0.522 + 0.476i)6-s + (−0.968 − 0.259i)7-s + (−0.249 + 0.249i)8-s + (0.418 + 0.908i)9-s + (−0.428 − 0.562i)10-s + (1.25 − 0.723i)11-s + (0.229 + 0.444i)12-s + (0.161 − 0.0432i)13-s + (−0.354 + 0.613i)14-s + (−0.941 + 0.335i)15-s + (0.125 + 0.216i)16-s + (1.06 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.337 + 0.941i$
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: \((2,\ 90,\ (\ :1/2),\ -0.337 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.489873 - 0.696020i\)
\(L(\frac12)\) \(\approx\) \(0.489873 - 0.696020i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.45 + 0.933i)T \)
5 \( 1 + (-1.36 + 1.76i)T \)
good7 \( 1 + (2.56 + 0.686i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-4.15 + 2.39i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.581 + 0.155i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-4.40 - 4.40i)T + 17iT^{2} \)
19 \( 1 - 5.19iT - 19T^{2} \)
23 \( 1 + (0.681 + 2.54i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (0.920 + 1.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.03 - 3.53i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.632 + 0.632i)T - 37iT^{2} \)
41 \( 1 + (5.58 + 3.22i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.644 - 2.40i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.02 - 3.82i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.31 - 1.31i)T - 53iT^{2} \)
59 \( 1 + (0.0645 - 0.111i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.27 - 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.85 - 10.6i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (-3.30 - 3.30i)T + 73iT^{2} \)
79 \( 1 + (-3.62 + 2.09i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.1 - 2.97i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 2.04T + 89T^{2} \)
97 \( 1 + (16.7 + 4.47i)T + (84.0 + 48.5i)T^{2} \)
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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

−13.45395208068434171326097529248, −12.56259507098898569706141552853, −11.98587078752174125145651426430, −10.57721676299724642075445703646, −9.702356688600856711364323784548, −8.349296288916503717921175974354, −6.41950928063750086463270343370, −5.63349037489866253487480040083, −3.84888966926430982850251889459, −1.33887093661092459659393263809, 3.49493859244497851959499834565, 5.20019686367190366723109706454, 6.41049696517324526275251756975, 7.05187253284389156503637721337, 9.428676507079804606162351085426, 9.724048168029027645579816043704, 11.28686055169635002520718968563, 12.30311595358981976039358895597, 13.53404614247401335249155382152, 14.65996249747862307233741246098

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