↑

Properties

Label 2-90-45.32-c1-0-0
Degree $2$
Conductor $90$
Sign $-0.328 - 0.944i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 − 0.258i)2-s + (−1.73 + 0.0795i)3-s + (0.866 + 0.499i)4-s + (−1.51 + 1.64i)5-s + (1.69 + 0.370i)6-s + (−1.00 + 3.75i)7-s + (−0.707 − 0.707i)8-s + (2.98 − 0.275i)9-s + (1.89 − 1.19i)10-s + (−3.44 + 1.98i)11-s + (−1.53 − 0.796i)12-s + (−0.256 − 0.956i)13-s + (1.94 − 3.36i)14-s + (2.49 − 2.95i)15-s + (0.500 + 0.866i)16-s + (0.120 − 0.120i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.998 + 0.0459i)3-s + (0.433 + 0.249i)4-s + (−0.679 + 0.733i)5-s + (0.690 + 0.151i)6-s + (−0.380 + 1.41i)7-s + (−0.249 − 0.249i)8-s + (0.995 − 0.0917i)9-s + (0.598 − 0.376i)10-s + (−1.03 + 0.599i)11-s + (−0.444 − 0.229i)12-s + (−0.0710 − 0.265i)13-s + (0.519 − 0.899i)14-s + (0.644 − 0.764i)15-s + (0.125 + 0.216i)16-s + (0.0291 − 0.0291i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
Sign: $-0.328 - 0.944i$
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: \((2,\ 90,\ (\ :1/2),\ -0.328 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.208610 + 0.293496i\)
\(L(\frac12)\) \(\approx\) \(0.208610 + 0.293496i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 0.0795i)T \)
5 \( 1 + (1.51 - 1.64i)T \)
good7 \( 1 + (1.00 - 3.75i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (3.44 - 1.98i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.256 + 0.956i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.120 + 0.120i)T - 17iT^{2} \)
19 \( 1 + 1.88iT - 19T^{2} \)
23 \( 1 + (-5.08 + 1.36i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.15 - 3.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.70 - 8.14i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.26 - 3.26i)T + 37iT^{2} \)
41 \( 1 + (-7.15 - 4.13i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.99 + 0.533i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-3.34 - 0.897i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.66 - 3.66i)T + 53iT^{2} \)
59 \( 1 + (2.72 - 4.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.35 + 7.54i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.86 + 2.10i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 6.94iT - 71T^{2} \)
73 \( 1 + (8.27 - 8.27i)T - 73iT^{2} \)
79 \( 1 + (-11.7 + 6.78i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.81 + 6.75i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 4.87T + 89T^{2} \)
97 \( 1 + (0.387 - 1.44i)T + (-84.0 - 48.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.96472441509450193790184907452, −12.79309291304006973236648691738, −12.18560624158737866481894388966, −11.10360773691629744994584167634, −10.34058356079336317102549784173, −9.049382547311596331398043764727, −7.59382230381229383766826271875, −6.50721842454296087605454564206, −5.09261905803672684379667766706, −2.81925653112444433162049505195, 0.61006501612829189267912984801, 4.11900219054840968756339190750, 5.61077309093021410393670229855, 7.13271460238210307692755659808, 7.921173575873059110492692729208, 9.536860956779032308503207368132, 10.66616038458908482950589325647, 11.34932792719517834616561593587, 12.68638118783345156022967675654, 13.51794754817995718839969461261

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