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Properties

Label 2-270-135.104-c2-0-11
Degree 22
Conductor 270270
Sign −0.588−0.808i-0.588 - 0.808i
Analytic cond. 7.356967.35696
Root an. cond. 2.712372.71237
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.32 + 0.483i)2-s + (−1.34 + 2.68i)3-s + (1.53 + 1.28i)4-s + (2.69 + 4.21i)5-s + (−3.08 + 2.91i)6-s + (2.97 + 3.54i)7-s + (1.41 + 2.44i)8-s + (−5.38 − 7.21i)9-s + (1.54 + 6.90i)10-s + (1.05 − 0.186i)11-s + (−5.50 + 2.38i)12-s + (1.47 + 4.06i)13-s + (2.23 + 6.14i)14-s + (−14.9 + 1.55i)15-s + (0.694 + 3.93i)16-s + (−1.77 + 3.07i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (−0.448 + 0.893i)3-s + (0.383 + 0.321i)4-s + (0.538 + 0.842i)5-s + (−0.513 + 0.485i)6-s + (0.424 + 0.505i)7-s + (0.176 + 0.306i)8-s + (−0.598 − 0.801i)9-s + (0.154 + 0.690i)10-s + (0.0959 − 0.0169i)11-s + (−0.458 + 0.198i)12-s + (0.113 + 0.312i)13-s + (0.159 + 0.438i)14-s + (−0.994 + 0.103i)15-s + (0.0434 + 0.246i)16-s + (−0.104 + 0.180i)17-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((−0.588−0.808i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1) L(s)=((−0.588−0.808i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: −0.588−0.808i-0.588 - 0.808i
Analytic conductor: 7.356967.35696
Root analytic conductor: 2.712372.71237
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ270(239,⋅)\chi_{270} (239, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 270, ( :1), −0.588−0.808i)(2,\ 270,\ (\ :1),\ -0.588 - 0.808i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.979890+1.92554i0.979890 + 1.92554i
L(12)L(\frac12) ≈\approx 0.979890+1.92554i0.979890 + 1.92554i
L(2)L(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+(−1.32−0.483i)T 1 + (-1.32 - 0.483i)T
3 1+(1.34−2.68i)T 1 + (1.34 - 2.68i)T
5 1+(−2.69−4.21i)T 1 + (-2.69 - 4.21i)T
good7 1+(−2.97−3.54i)T+(−8.50+48.2i)T2 1 + (-2.97 - 3.54i)T + (-8.50 + 48.2i)T^{2}
11 1+(−1.05+0.186i)T+(113.−41.3i)T2 1 + (-1.05 + 0.186i)T + (113. - 41.3i)T^{2}
13 1+(−1.47−4.06i)T+(−129.+108.i)T2 1 + (-1.47 - 4.06i)T + (-129. + 108. i)T^{2}
17 1+(1.77−3.07i)T+(−144.5−250.i)T2 1 + (1.77 - 3.07i)T + (-144.5 - 250. i)T^{2}
19 1+(1.99+3.45i)T+(−180.5+312.i)T2 1 + (1.99 + 3.45i)T + (-180.5 + 312. i)T^{2}
23 1+(19.8+16.6i)T+(91.8+520.i)T2 1 + (19.8 + 16.6i)T + (91.8 + 520. i)T^{2}
29 1+(2.24−6.16i)T+(−644.−540.i)T2 1 + (2.24 - 6.16i)T + (-644. - 540. i)T^{2}
31 1+(11.5+9.67i)T+(166.+946.i)T2 1 + (11.5 + 9.67i)T + (166. + 946. i)T^{2}
37 1+(−63.7−36.7i)T+(684.5+1.18e3i)T2 1 + (-63.7 - 36.7i)T + (684.5 + 1.18e3i)T^{2}
41 1+(−6.76−18.5i)T+(−1.28e3+1.08e3i)T2 1 + (-6.76 - 18.5i)T + (-1.28e3 + 1.08e3i)T^{2}
43 1+(4.11−0.725i)T+(1.73e3−632.i)T2 1 + (4.11 - 0.725i)T + (1.73e3 - 632. i)T^{2}
47 1+(−18.0+15.1i)T+(383.−2.17e3i)T2 1 + (-18.0 + 15.1i)T + (383. - 2.17e3i)T^{2}
53 1+20.2T+2.80e3T2 1 + 20.2T + 2.80e3T^{2}
59 1+(−55.8−9.85i)T+(3.27e3+1.19e3i)T2 1 + (-55.8 - 9.85i)T + (3.27e3 + 1.19e3i)T^{2}
61 1+(−76.1+63.8i)T+(646.−3.66e3i)T2 1 + (-76.1 + 63.8i)T + (646. - 3.66e3i)T^{2}
67 1+(6.06+16.6i)T+(−3.43e3+2.88e3i)T2 1 + (6.06 + 16.6i)T + (-3.43e3 + 2.88e3i)T^{2}
71 1+(26.1+15.1i)T+(2.52e3+4.36e3i)T2 1 + (26.1 + 15.1i)T + (2.52e3 + 4.36e3i)T^{2}
73 1+(−83.2+48.0i)T+(2.66e3−4.61e3i)T2 1 + (-83.2 + 48.0i)T + (2.66e3 - 4.61e3i)T^{2}
79 1+(−132.−48.2i)T+(4.78e3+4.01e3i)T2 1 + (-132. - 48.2i)T + (4.78e3 + 4.01e3i)T^{2}
83 1+(107.+39.2i)T+(5.27e3+4.42e3i)T2 1 + (107. + 39.2i)T + (5.27e3 + 4.42e3i)T^{2}
89 1+(−42.0+24.2i)T+(3.96e3−6.85e3i)T2 1 + (-42.0 + 24.2i)T + (3.96e3 - 6.85e3i)T^{2}
97 1+(−61.5+10.8i)T+(8.84e3−3.21e3i)T2 1 + (-61.5 + 10.8i)T + (8.84e3 - 3.21e3i)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

−11.79931340804102052780068655306, −11.20550186703949631714089296120, −10.28562189069756854788063286373, −9.347566196976437441261057966538, −8.142001838941328935828478529504, −6.64478036429519282603456595060, −5.95381267358404229980739075933, −4.90992355159999401567312311172, −3.75145821602763427736072719045, −2.39603007216253393251957406268, 0.983991680417967870014835976354, 2.22593775174422230376861169296, 4.15662836306999486406616024900, 5.34713598711390483596358511173, 6.06995579909803390200240686407, 7.33301950917654847076470163620, 8.255933274539144969999214653309, 9.582321824390457579840084059733, 10.72560483282889038216265787111, 11.57809716850877707967993943368

Graph of the ZZ-function along the critical line