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Properties

Label 2-538-269.82-c2-0-14
Degree 22
Conductor 538538
Sign −0.00709−0.999i-0.00709 - 0.999i
Analytic cond. 14.659414.6594
Root an. cond. 3.828763.82876
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 + i)2-s + (1.35 − 1.35i)3-s − 2i·4-s + 4.13·5-s + 2.70i·6-s + (1.20 + 1.20i)7-s + (2 + 2i)8-s + 5.32i·9-s + (−4.13 + 4.13i)10-s + 19.3i·11-s + (−2.70 − 2.70i)12-s − 6.76i·13-s − 2.41·14-s + (5.60 − 5.60i)15-s − 4·16-s + (−16.6 + 16.6i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.451 − 0.451i)3-s − 0.5i·4-s + 0.826·5-s + 0.451i·6-s + (0.172 + 0.172i)7-s + (0.250 + 0.250i)8-s + 0.592i·9-s + (−0.413 + 0.413i)10-s + 1.75i·11-s + (−0.225 − 0.225i)12-s − 0.520i·13-s − 0.172·14-s + (0.373 − 0.373i)15-s − 0.250·16-s + (−0.980 + 0.980i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((−0.00709−0.999i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00709 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1) L(s)=((−0.00709−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.00709 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: −0.00709−0.999i-0.00709 - 0.999i
Analytic conductor: 14.659414.6594
Root analytic conductor: 3.828763.82876
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ538(351,⋅)\chi_{538} (351, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 538, ( :1), −0.00709−0.999i)(2,\ 538,\ (\ :1),\ -0.00709 - 0.999i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.6132206501.613220650
L(12)L(\frac12) ≈\approx 1.6132206501.613220650
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−i)T 1 + (1 - i)T
269 1+(−242.−117.i)T 1 + (-242. - 117. i)T
good3 1+(−1.35+1.35i)T−9iT2 1 + (-1.35 + 1.35i)T - 9iT^{2}
5 1−4.13T+25T2 1 - 4.13T + 25T^{2}
7 1+(−1.20−1.20i)T+49iT2 1 + (-1.20 - 1.20i)T + 49iT^{2}
11 1−19.3iT−121T2 1 - 19.3iT - 121T^{2}
13 1+6.76iT−169T2 1 + 6.76iT - 169T^{2}
17 1+(16.6−16.6i)T−289iT2 1 + (16.6 - 16.6i)T - 289iT^{2}
19 1+(−1.42−1.42i)T+361iT2 1 + (-1.42 - 1.42i)T + 361iT^{2}
23 1+16.5T+529T2 1 + 16.5T + 529T^{2}
29 1+(−15.5−15.5i)T+841iT2 1 + (-15.5 - 15.5i)T + 841iT^{2}
31 1+(4.83+4.83i)T+961iT2 1 + (4.83 + 4.83i)T + 961iT^{2}
37 1−70.2T+1.36e3T2 1 - 70.2T + 1.36e3T^{2}
41 1+70.2T+1.68e3T2 1 + 70.2T + 1.68e3T^{2}
43 1−8.53iT−1.84e3T2 1 - 8.53iT - 1.84e3T^{2}
47 1−42.1T+2.20e3T2 1 - 42.1T + 2.20e3T^{2}
53 1+14.6T+2.80e3T2 1 + 14.6T + 2.80e3T^{2}
59 1+(−73.5−73.5i)T+3.48e3iT2 1 + (-73.5 - 73.5i)T + 3.48e3iT^{2}
61 1−9.57T+3.72e3T2 1 - 9.57T + 3.72e3T^{2}
67 1−46.8T+4.48e3T2 1 - 46.8T + 4.48e3T^{2}
71 1+(55.1+55.1i)T+5.04e3iT2 1 + (55.1 + 55.1i)T + 5.04e3iT^{2}
73 1−140.iT−5.32e3T2 1 - 140. iT - 5.32e3T^{2}
79 1+138.iT−6.24e3T2 1 + 138. iT - 6.24e3T^{2}
83 1+(31.5+31.5i)T+6.88e3iT2 1 + (31.5 + 31.5i)T + 6.88e3iT^{2}
89 1−90.2iT−7.92e3T2 1 - 90.2iT - 7.92e3T^{2}
97 1−70.9iT−9.40e3T2 1 - 70.9iT - 9.40e3T^{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

−10.36383279682284346408226401754, −10.01426704443752341005882333466, −8.940071359209233196319027526329, −8.125095800053820195640163983413, −7.33683866355807811918215378748, −6.45296453024541436511604139830, −5.41547860753927204733200502883, −4.38191023923130499142904228412, −2.35004758233802669134358060412, −1.70795927638412764206993037959, 0.69214441311686467217315960465, 2.30977433649205079342743479340, 3.34942442999658786577398306506, 4.42565541050867798601581146948, 5.86464113482176289372654504239, 6.70144601824522740602123289860, 8.100149263160922657960629867512, 8.860556626934470182907160230027, 9.493071738832776871235856621564, 10.20314143669211838358534982796

Graph of the ZZ-function along the critical line