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Properties

Label 2-270-135.122-c1-0-7
Degree 22
Conductor 270270
Sign 0.654−0.756i0.654 - 0.756i
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 + (−1.72 + 0.169i)3-s + (0.342 + 0.939i)4-s + (2.07 − 0.845i)5-s + (−1.50 − 0.849i)6-s + (0.101 + 0.217i)7-s + (−0.258 + 0.965i)8-s + (2.94 − 0.585i)9-s + (2.18 + 0.494i)10-s + (2.07 + 2.47i)11-s + (−0.749 − 1.56i)12-s + (1.10 + 1.57i)13-s + (−0.0417 + 0.236i)14-s + (−3.42 + 1.80i)15-s + (−0.766 + 0.642i)16-s + (0.935 + 3.48i)17-s + ⋯
L(s)  = 1  + (0.579 + 0.405i)2-s + (−0.995 + 0.0980i)3-s + (0.171 + 0.469i)4-s + (0.925 − 0.378i)5-s + (−0.616 − 0.346i)6-s + (0.0383 + 0.0823i)7-s + (−0.0915 + 0.341i)8-s + (0.980 − 0.195i)9-s + (0.689 + 0.156i)10-s + (0.625 + 0.745i)11-s + (−0.216 − 0.450i)12-s + (0.306 + 0.437i)13-s + (−0.0111 + 0.0632i)14-s + (−0.884 + 0.466i)15-s + (−0.191 + 0.160i)16-s + (0.226 + 0.846i)17-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((0.654−0.756i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1/2) L(s)=((0.654−0.756i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: 0.654−0.756i0.654 - 0.756i
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), 0.654−0.756i)(2,\ 270,\ (\ :1/2),\ 0.654 - 0.756i)

Particular Values

L(1)L(1) ≈\approx 1.39133+0.636105i1.39133 + 0.636105i
L(12)L(\frac12) ≈\approx 1.39133+0.636105i1.39133 + 0.636105i
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+(1.72−0.169i)T 1 + (1.72 - 0.169i)T
5 1+(−2.07+0.845i)T 1 + (-2.07 + 0.845i)T
good7 1+(−0.101−0.217i)T+(−4.49+5.36i)T2 1 + (-0.101 - 0.217i)T + (-4.49 + 5.36i)T^{2}
11 1+(−2.07−2.47i)T+(−1.91+10.8i)T2 1 + (-2.07 - 2.47i)T + (-1.91 + 10.8i)T^{2}
13 1+(−1.10−1.57i)T+(−4.44+12.2i)T2 1 + (-1.10 - 1.57i)T + (-4.44 + 12.2i)T^{2}
17 1+(−0.935−3.48i)T+(−14.7+8.5i)T2 1 + (-0.935 - 3.48i)T + (-14.7 + 8.5i)T^{2}
19 1+(−1.18+0.683i)T+(9.5−16.4i)T2 1 + (-1.18 + 0.683i)T + (9.5 - 16.4i)T^{2}
23 1+(6.16+2.87i)T+(14.7+17.6i)T2 1 + (6.16 + 2.87i)T + (14.7 + 17.6i)T^{2}
29 1+(0.340+1.92i)T+(−27.2+9.91i)T2 1 + (0.340 + 1.92i)T + (-27.2 + 9.91i)T^{2}
31 1+(−2.42+0.881i)T+(23.7−19.9i)T2 1 + (-2.42 + 0.881i)T + (23.7 - 19.9i)T^{2}
37 1+(−5.53+1.48i)T+(32.0−18.5i)T2 1 + (-5.53 + 1.48i)T + (32.0 - 18.5i)T^{2}
41 1+(−0.258−0.0455i)T+(38.5+14.0i)T2 1 + (-0.258 - 0.0455i)T + (38.5 + 14.0i)T^{2}
43 1+(−0.573+6.54i)T+(−42.3−7.46i)T2 1 + (-0.573 + 6.54i)T + (-42.3 - 7.46i)T^{2}
47 1+(11.9−5.56i)T+(30.2−36.0i)T2 1 + (11.9 - 5.56i)T + (30.2 - 36.0i)T^{2}
53 1+(9.15+9.15i)T+53iT2 1 + (9.15 + 9.15i)T + 53iT^{2}
59 1+(4.98+4.18i)T+(10.2+58.1i)T2 1 + (4.98 + 4.18i)T + (10.2 + 58.1i)T^{2}
61 1+(14.0+5.13i)T+(46.7+39.2i)T2 1 + (14.0 + 5.13i)T + (46.7 + 39.2i)T^{2}
67 1+(3.34−2.34i)T+(22.9−62.9i)T2 1 + (3.34 - 2.34i)T + (22.9 - 62.9i)T^{2}
71 1+(−9.56−5.52i)T+(35.5+61.4i)T2 1 + (-9.56 - 5.52i)T + (35.5 + 61.4i)T^{2}
73 1+(−2.32−0.623i)T+(63.2+36.5i)T2 1 + (-2.32 - 0.623i)T + (63.2 + 36.5i)T^{2}
79 1+(−2.06+0.364i)T+(74.2−27.0i)T2 1 + (-2.06 + 0.364i)T + (74.2 - 27.0i)T^{2}
83 1+(5.39−7.69i)T+(−28.3−77.9i)T2 1 + (5.39 - 7.69i)T + (-28.3 - 77.9i)T^{2}
89 1+(−1.90−3.29i)T+(−44.5+77.0i)T2 1 + (-1.90 - 3.29i)T + (-44.5 + 77.0i)T^{2}
97 1+(7.17+0.627i)T+(95.5+16.8i)T2 1 + (7.17 + 0.627i)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

−12.24866534377630973784502385920, −11.29896161030175588103045108700, −10.14765608870265169714046018071, −9.383809842216087206251988242754, −8.015841814489051551796170786622, −6.58541700135475499592437965394, −6.10789813974056648795115349318, −4.98354528917355169643562673041, −4.05431294458814606930417169524, −1.81554551080132988763969218777, 1.39677677731163618377617472832, 3.17570090883762488326061285671, 4.68520498058350383825984503296, 5.85446256013743660967492428383, 6.30434028527976866861863285478, 7.61963301760474160442486285135, 9.350813243192121706065566330353, 10.13808201695676951396321565473, 11.02839051113452761582959044866, 11.71090928263355561365852147114

Graph of the ZZ-function along the critical line