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Properties

Label 2-270-135.122-c1-0-4
Degree 22
Conductor 270270
Sign −0.541−0.840i-0.541 - 0.840i
Analytic cond. 2.155962.15596
Root an. cond. 1.468311.46831
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.819 + 0.573i)2-s + (0.264 + 1.71i)3-s + (0.342 + 0.939i)4-s + (−2.23 − 0.138i)5-s + (−0.765 + 1.55i)6-s + (0.660 + 1.41i)7-s + (−0.258 + 0.965i)8-s + (−2.86 + 0.903i)9-s + (−1.74 − 1.39i)10-s + (2.49 + 2.97i)11-s + (−1.51 + 0.833i)12-s + (−2.53 − 3.62i)13-s + (−0.271 + 1.53i)14-s + (−0.352 − 3.85i)15-s + (−0.766 + 0.642i)16-s + (1.52 + 5.69i)17-s + ⋯
L(s)  = 1  + (0.579 + 0.405i)2-s + (0.152 + 0.988i)3-s + (0.171 + 0.469i)4-s + (−0.998 − 0.0618i)5-s + (−0.312 + 0.634i)6-s + (0.249 + 0.535i)7-s + (−0.0915 + 0.341i)8-s + (−0.953 + 0.301i)9-s + (−0.553 − 0.440i)10-s + (0.753 + 0.897i)11-s + (−0.438 + 0.240i)12-s + (−0.703 − 1.00i)13-s + (−0.0725 + 0.411i)14-s + (−0.0910 − 0.995i)15-s + (−0.191 + 0.160i)16-s + (0.370 + 1.38i)17-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((−0.541−0.840i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1/2) L(s)=((−0.541−0.840i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: −0.541−0.840i-0.541 - 0.840i
Analytic conductor: 2.155962.15596
Root analytic conductor: 1.468311.46831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ270(257,⋅)\chi_{270} (257, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 270, ( :1/2), −0.541−0.840i)(2,\ 270,\ (\ :1/2),\ -0.541 - 0.840i)

Particular Values

L(1)L(1) ≈\approx 0.726040+1.33188i0.726040 + 1.33188i
L(12)L(\frac12) ≈\approx 0.726040+1.33188i0.726040 + 1.33188i
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+(−0.819−0.573i)T 1 + (-0.819 - 0.573i)T
3 1+(−0.264−1.71i)T 1 + (-0.264 - 1.71i)T
5 1+(2.23+0.138i)T 1 + (2.23 + 0.138i)T
good7 1+(−0.660−1.41i)T+(−4.49+5.36i)T2 1 + (-0.660 - 1.41i)T + (-4.49 + 5.36i)T^{2}
11 1+(−2.49−2.97i)T+(−1.91+10.8i)T2 1 + (-2.49 - 2.97i)T + (-1.91 + 10.8i)T^{2}
13 1+(2.53+3.62i)T+(−4.44+12.2i)T2 1 + (2.53 + 3.62i)T + (-4.44 + 12.2i)T^{2}
17 1+(−1.52−5.69i)T+(−14.7+8.5i)T2 1 + (-1.52 - 5.69i)T + (-14.7 + 8.5i)T^{2}
19 1+(−5.43+3.13i)T+(9.5−16.4i)T2 1 + (-5.43 + 3.13i)T + (9.5 - 16.4i)T^{2}
23 1+(1.84+0.862i)T+(14.7+17.6i)T2 1 + (1.84 + 0.862i)T + (14.7 + 17.6i)T^{2}
29 1+(0.651+3.69i)T+(−27.2+9.91i)T2 1 + (0.651 + 3.69i)T + (-27.2 + 9.91i)T^{2}
31 1+(−4.48+1.63i)T+(23.7−19.9i)T2 1 + (-4.48 + 1.63i)T + (23.7 - 19.9i)T^{2}
37 1+(−5.27+1.41i)T+(32.0−18.5i)T2 1 + (-5.27 + 1.41i)T + (32.0 - 18.5i)T^{2}
41 1+(−1.36−0.240i)T+(38.5+14.0i)T2 1 + (-1.36 - 0.240i)T + (38.5 + 14.0i)T^{2}
43 1+(0.741−8.47i)T+(−42.3−7.46i)T2 1 + (0.741 - 8.47i)T + (-42.3 - 7.46i)T^{2}
47 1+(2.58−1.20i)T+(30.2−36.0i)T2 1 + (2.58 - 1.20i)T + (30.2 - 36.0i)T^{2}
53 1+(−4.58−4.58i)T+53iT2 1 + (-4.58 - 4.58i)T + 53iT^{2}
59 1+(−0.286−0.240i)T+(10.2+58.1i)T2 1 + (-0.286 - 0.240i)T + (10.2 + 58.1i)T^{2}
61 1+(−3.34−1.21i)T+(46.7+39.2i)T2 1 + (-3.34 - 1.21i)T + (46.7 + 39.2i)T^{2}
67 1+(−9.14+6.40i)T+(22.9−62.9i)T2 1 + (-9.14 + 6.40i)T + (22.9 - 62.9i)T^{2}
71 1+(13.8+7.99i)T+(35.5+61.4i)T2 1 + (13.8 + 7.99i)T + (35.5 + 61.4i)T^{2}
73 1+(15.8+4.24i)T+(63.2+36.5i)T2 1 + (15.8 + 4.24i)T + (63.2 + 36.5i)T^{2}
79 1+(−0.515+0.0908i)T+(74.2−27.0i)T2 1 + (-0.515 + 0.0908i)T + (74.2 - 27.0i)T^{2}
83 1+(−6.06+8.66i)T+(−28.3−77.9i)T2 1 + (-6.06 + 8.66i)T + (-28.3 - 77.9i)T^{2}
89 1+(2.59+4.49i)T+(−44.5+77.0i)T2 1 + (2.59 + 4.49i)T + (-44.5 + 77.0i)T^{2}
97 1+(11.4+1.00i)T+(95.5+16.8i)T2 1 + (11.4 + 1.00i)T + (95.5 + 16.8i)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

−12.07016996841701875619584619138, −11.57754615700751342118738579815, −10.35686731877249381868935404615, −9.360211614288587877751584808687, −8.228096452384737037614173885936, −7.53300667685356143112092775831, −6.00467621767809857972921601858, −4.87752262758142254624901490751, −4.07701804646511460352838907064, −2.88015023871391165233455174295, 1.06568391578984764474728369134, 2.93648337889104611703500157957, 4.03617933401788104034199485399, 5.42103167874383518748810763811, 6.85088725814550800218979919673, 7.42585290291932953146723395861, 8.570832259285214721706850205714, 9.733781267601093933148486188033, 11.23479946185670083747295597405, 11.77797577196894023887109453869

Graph of the ZZ-function along the critical line