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Properties

Label 2-90-45.32-c1-0-0
Degree 22
Conductor 9090
Sign −0.328−0.944i-0.328 - 0.944i
Analytic cond. 0.7186530.718653
Root an. cond. 0.8477340.847734
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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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

Λ(s)=(90s/2 ΓC(s) L(s)=((−0.328−0.944i) Λ‾(2−s)\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}
Λ(s)=(90s/2 ΓC(s+1/2) L(s)=((−0.328−0.944i) Λ‾(1−s)\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: 22
Conductor: 9090    =    2⋅32⋅52 \cdot 3^{2} \cdot 5
Sign: −0.328−0.944i-0.328 - 0.944i
Analytic conductor: 0.7186530.718653
Root analytic conductor: 0.8477340.847734
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ90(77,⋅)\chi_{90} (77, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 90, ( :1/2), −0.328−0.944i)(2,\ 90,\ (\ :1/2),\ -0.328 - 0.944i)

Particular Values

L(1)L(1) ≈\approx 0.208610+0.293496i0.208610 + 0.293496i
L(12)L(\frac12) ≈\approx 0.208610+0.293496i0.208610 + 0.293496i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.965+0.258i)T 1 + (0.965 + 0.258i)T
3 1+(1.73−0.0795i)T 1 + (1.73 - 0.0795i)T
5 1+(1.51−1.64i)T 1 + (1.51 - 1.64i)T
good7 1+(1.00−3.75i)T+(−6.06−3.5i)T2 1 + (1.00 - 3.75i)T + (-6.06 - 3.5i)T^{2}
11 1+(3.44−1.98i)T+(5.5−9.52i)T2 1 + (3.44 - 1.98i)T + (5.5 - 9.52i)T^{2}
13 1+(0.256+0.956i)T+(−11.2+6.5i)T2 1 + (0.256 + 0.956i)T + (-11.2 + 6.5i)T^{2}
17 1+(−0.120+0.120i)T−17iT2 1 + (-0.120 + 0.120i)T - 17iT^{2}
19 1+1.88iT−19T2 1 + 1.88iT - 19T^{2}
23 1+(−5.08+1.36i)T+(19.9−11.5i)T2 1 + (-5.08 + 1.36i)T + (19.9 - 11.5i)T^{2}
29 1+(−2.15−3.73i)T+(−14.5+25.1i)T2 1 + (-2.15 - 3.73i)T + (-14.5 + 25.1i)T^{2}
31 1+(4.70−8.14i)T+(−15.5−26.8i)T2 1 + (4.70 - 8.14i)T + (-15.5 - 26.8i)T^{2}
37 1+(−3.26−3.26i)T+37iT2 1 + (-3.26 - 3.26i)T + 37iT^{2}
41 1+(−7.15−4.13i)T+(20.5+35.5i)T2 1 + (-7.15 - 4.13i)T + (20.5 + 35.5i)T^{2}
43 1+(1.99+0.533i)T+(37.2+21.5i)T2 1 + (1.99 + 0.533i)T + (37.2 + 21.5i)T^{2}
47 1+(−3.34−0.897i)T+(40.7+23.5i)T2 1 + (-3.34 - 0.897i)T + (40.7 + 23.5i)T^{2}
53 1+(−3.66−3.66i)T+53iT2 1 + (-3.66 - 3.66i)T + 53iT^{2}
59 1+(2.72−4.72i)T+(−29.5−51.0i)T2 1 + (2.72 - 4.72i)T + (-29.5 - 51.0i)T^{2}
61 1+(4.35+7.54i)T+(−30.5+52.8i)T2 1 + (4.35 + 7.54i)T + (-30.5 + 52.8i)T^{2}
67 1+(−7.86+2.10i)T+(58.0−33.5i)T2 1 + (-7.86 + 2.10i)T + (58.0 - 33.5i)T^{2}
71 1+6.94iT−71T2 1 + 6.94iT - 71T^{2}
73 1+(8.27−8.27i)T−73iT2 1 + (8.27 - 8.27i)T - 73iT^{2}
79 1+(−11.7+6.78i)T+(39.5−68.4i)T2 1 + (-11.7 + 6.78i)T + (39.5 - 68.4i)T^{2}
83 1+(−1.81+6.75i)T+(−71.8−41.5i)T2 1 + (-1.81 + 6.75i)T + (-71.8 - 41.5i)T^{2}
89 1+4.87T+89T2 1 + 4.87T + 89T^{2}
97 1+(0.387−1.44i)T+(−84.0−48.5i)T2 1 + (0.387 - 1.44i)T + (-84.0 - 48.5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line