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Properties

Label 2-33-33.2-c3-0-4
Degree 22
Conductor 3333
Sign 0.468−0.883i0.468 - 0.883i
Analytic cond. 1.947061.94706
Root an. cond. 1.395371.39537
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.882 + 2.71i)2-s + (4.90 − 1.71i)3-s + (−0.124 − 0.0905i)4-s + (4.91 − 1.59i)5-s + (0.325 + 14.8i)6-s + (−9.45 + 13.0i)7-s + (−18.1 + 13.1i)8-s + (21.1 − 16.8i)9-s + 14.7i·10-s + (−5.09 − 36.1i)11-s + (−0.766 − 0.230i)12-s + (16.5 + 5.38i)13-s + (−26.9 − 37.1i)14-s + (21.3 − 16.2i)15-s + (−20.1 − 62.0i)16-s + (−30.8 − 94.8i)17-s + ⋯
L(s)  = 1  + (−0.311 + 0.960i)2-s + (0.944 − 0.329i)3-s + (−0.0155 − 0.0113i)4-s + (0.439 − 0.142i)5-s + (0.0221 + 1.00i)6-s + (−0.510 + 0.702i)7-s + (−0.801 + 0.581i)8-s + (0.782 − 0.622i)9-s + 0.466i·10-s + (−0.139 − 0.990i)11-s + (−0.0184 − 0.00554i)12-s + (0.353 + 0.114i)13-s + (−0.515 − 0.709i)14-s + (0.368 − 0.279i)15-s + (−0.314 − 0.969i)16-s + (−0.439 − 1.35i)17-s + ⋯

Functional equation

Λ(s)=(33s/2 ΓC(s) L(s)=((0.468−0.883i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(33s/2 ΓC(s+3/2) L(s)=((0.468−0.883i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3333    =    3⋅113 \cdot 11
Sign: 0.468−0.883i0.468 - 0.883i
Analytic conductor: 1.947061.94706
Root analytic conductor: 1.395371.39537
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ33(2,⋅)\chi_{33} (2, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 33, ( :3/2), 0.468−0.883i)(2,\ 33,\ (\ :3/2),\ 0.468 - 0.883i)

Particular Values

L(2)L(2) ≈\approx 1.23933+0.745153i1.23933 + 0.745153i
L(12)L(\frac12) ≈\approx 1.23933+0.745153i1.23933 + 0.745153i
L(52)L(\frac{5}{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)
bad3 1+(−4.90+1.71i)T 1 + (-4.90 + 1.71i)T
11 1+(5.09+36.1i)T 1 + (5.09 + 36.1i)T
good2 1+(0.882−2.71i)T+(−6.47−4.70i)T2 1 + (0.882 - 2.71i)T + (-6.47 - 4.70i)T^{2}
5 1+(−4.91+1.59i)T+(101.−73.4i)T2 1 + (-4.91 + 1.59i)T + (101. - 73.4i)T^{2}
7 1+(9.45−13.0i)T+(−105.−326.i)T2 1 + (9.45 - 13.0i)T + (-105. - 326. i)T^{2}
13 1+(−16.5−5.38i)T+(1.77e3+1.29e3i)T2 1 + (-16.5 - 5.38i)T + (1.77e3 + 1.29e3i)T^{2}
17 1+(30.8+94.8i)T+(−3.97e3+2.88e3i)T2 1 + (30.8 + 94.8i)T + (-3.97e3 + 2.88e3i)T^{2}
19 1+(27.1+37.3i)T+(−2.11e3+6.52e3i)T2 1 + (27.1 + 37.3i)T + (-2.11e3 + 6.52e3i)T^{2}
23 1−113.iT−1.21e4T2 1 - 113. iT - 1.21e4T^{2}
29 1+(−110.−80.2i)T+(7.53e3+2.31e4i)T2 1 + (-110. - 80.2i)T + (7.53e3 + 2.31e4i)T^{2}
31 1+(−26.9+82.9i)T+(−2.41e4−1.75e4i)T2 1 + (-26.9 + 82.9i)T + (-2.41e4 - 1.75e4i)T^{2}
37 1+(216.+157.i)T+(1.56e4+4.81e4i)T2 1 + (216. + 157. i)T + (1.56e4 + 4.81e4i)T^{2}
41 1+(−21.4+15.5i)T+(2.12e4−6.55e4i)T2 1 + (-21.4 + 15.5i)T + (2.12e4 - 6.55e4i)T^{2}
43 1−487.iT−7.95e4T2 1 - 487. iT - 7.95e4T^{2}
47 1+(−35.3−48.6i)T+(−3.20e4+9.87e4i)T2 1 + (-35.3 - 48.6i)T + (-3.20e4 + 9.87e4i)T^{2}
53 1+(−472.−153.i)T+(1.20e5+8.75e4i)T2 1 + (-472. - 153. i)T + (1.20e5 + 8.75e4i)T^{2}
59 1+(432.−594.i)T+(−6.34e4−1.95e5i)T2 1 + (432. - 594. i)T + (-6.34e4 - 1.95e5i)T^{2}
61 1+(−817.+265.i)T+(1.83e5−1.33e5i)T2 1 + (-817. + 265. i)T + (1.83e5 - 1.33e5i)T^{2}
67 1−507.T+3.00e5T2 1 - 507.T + 3.00e5T^{2}
71 1+(980.−318.i)T+(2.89e5−2.10e5i)T2 1 + (980. - 318. i)T + (2.89e5 - 2.10e5i)T^{2}
73 1+(173.−238.i)T+(−1.20e5−3.69e5i)T2 1 + (173. - 238. i)T + (-1.20e5 - 3.69e5i)T^{2}
79 1+(576.+187.i)T+(3.98e5+2.89e5i)T2 1 + (576. + 187. i)T + (3.98e5 + 2.89e5i)T^{2}
83 1+(−39.9−122.i)T+(−4.62e5+3.36e5i)T2 1 + (-39.9 - 122. i)T + (-4.62e5 + 3.36e5i)T^{2}
89 1+811.iT−7.04e5T2 1 + 811. iT - 7.04e5T^{2}
97 1+(−69.6+214.i)T+(−7.38e5−5.36e5i)T2 1 + (-69.6 + 214. i)T + (-7.38e5 - 5.36e5i)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

−16.08845653053080266510311356188, −15.53526069369978445723398630798, −14.15524596708766301235918862222, −13.17101724773071223903801633166, −11.63296376826875181684806503581, −9.377999972402967986415012163626, −8.611481372456702636288596885177, −7.19650692967345661802187891516, −5.86777798127923540076194789436, −2.84341583733368694235431141278, 2.09925376066474982606005748051, 3.88069030332344983264937873919, 6.67402340235424028372737839532, 8.581826217731287210083299762417, 10.17051398608881263470064274947, 10.34466981954524859079962270063, 12.39235967277140646818414853011, 13.46130146427127657645952900619, 14.81736892621768921957645459917, 15.82520053480503653290614241292

Graph of the ZZ-function along the critical line