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Properties

Label 2-2e4-16.13-c5-0-0
Degree 22
Conductor 1616
Sign −0.438−0.898i-0.438 - 0.898i
Analytic cond. 2.566142.56614
Root an. cond. 1.601911.60191
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.25 − 2.08i)2-s + (−3.18 − 3.18i)3-s + (23.3 + 21.9i)4-s + (−67.3 + 67.3i)5-s + (10.0 + 23.3i)6-s + 148. i·7-s + (−76.8 − 163. i)8-s − 222. i·9-s + (494. − 213. i)10-s + (−256. + 256. i)11-s + (−4.40 − 143. i)12-s + (−218. − 218. i)13-s + (309. − 780. i)14-s + 428.·15-s + (62.6 + 1.02e3i)16-s − 463.·17-s + ⋯
L(s)  = 1  + (−0.929 − 0.368i)2-s + (−0.203 − 0.203i)3-s + (0.728 + 0.685i)4-s + (−1.20 + 1.20i)5-s + (0.114 + 0.264i)6-s + 1.14i·7-s + (−0.424 − 0.905i)8-s − 0.916i·9-s + (1.56 − 0.676i)10-s + (−0.638 + 0.638i)11-s + (−0.00883 − 0.288i)12-s + (−0.358 − 0.358i)13-s + (0.421 − 1.06i)14-s + 0.491·15-s + (0.0612 + 0.998i)16-s − 0.388·17-s + ⋯

Functional equation

Λ(s)=(16s/2 ΓC(s) L(s)=((−0.438−0.898i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(16s/2 ΓC(s+5/2) L(s)=((−0.438−0.898i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1616    =    242^{4}
Sign: −0.438−0.898i-0.438 - 0.898i
Analytic conductor: 2.566142.56614
Root analytic conductor: 1.601911.60191
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ16(13,⋅)\chi_{16} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 16, ( :5/2), −0.438−0.898i)(2,\ 16,\ (\ :5/2),\ -0.438 - 0.898i)

Particular Values

L(3)L(3) ≈\approx 0.210995+0.337730i0.210995 + 0.337730i
L(12)L(\frac12) ≈\approx 0.210995+0.337730i0.210995 + 0.337730i
L(72)L(\frac{7}{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+(5.25+2.08i)T 1 + (5.25 + 2.08i)T
good3 1+(3.18+3.18i)T+243iT2 1 + (3.18 + 3.18i)T + 243iT^{2}
5 1+(67.3−67.3i)T−3.12e3iT2 1 + (67.3 - 67.3i)T - 3.12e3iT^{2}
7 1−148.iT−1.68e4T2 1 - 148. iT - 1.68e4T^{2}
11 1+(256.−256.i)T−1.61e5iT2 1 + (256. - 256. i)T - 1.61e5iT^{2}
13 1+(218.+218.i)T+3.71e5iT2 1 + (218. + 218. i)T + 3.71e5iT^{2}
17 1+463.T+1.41e6T2 1 + 463.T + 1.41e6T^{2}
19 1+(−920.−920.i)T+2.47e6iT2 1 + (-920. - 920. i)T + 2.47e6iT^{2}
23 1−1.05e3iT−6.43e6T2 1 - 1.05e3iT - 6.43e6T^{2}
29 1+(−1.29e3−1.29e3i)T+2.05e7iT2 1 + (-1.29e3 - 1.29e3i)T + 2.05e7iT^{2}
31 1−1.00e4T+2.86e7T2 1 - 1.00e4T + 2.86e7T^{2}
37 1+(9.40e3−9.40e3i)T−6.93e7iT2 1 + (9.40e3 - 9.40e3i)T - 6.93e7iT^{2}
41 1−368.iT−1.15e8T2 1 - 368. iT - 1.15e8T^{2}
43 1+(9.16e3−9.16e3i)T−1.47e8iT2 1 + (9.16e3 - 9.16e3i)T - 1.47e8iT^{2}
47 1+7.63e3T+2.29e8T2 1 + 7.63e3T + 2.29e8T^{2}
53 1+(1.24e3−1.24e3i)T−4.18e8iT2 1 + (1.24e3 - 1.24e3i)T - 4.18e8iT^{2}
59 1+(−1.62e4+1.62e4i)T−7.14e8iT2 1 + (-1.62e4 + 1.62e4i)T - 7.14e8iT^{2}
61 1+(1.39e3+1.39e3i)T+8.44e8iT2 1 + (1.39e3 + 1.39e3i)T + 8.44e8iT^{2}
67 1+(1.74e4+1.74e4i)T+1.35e9iT2 1 + (1.74e4 + 1.74e4i)T + 1.35e9iT^{2}
71 1−6.74e4iT−1.80e9T2 1 - 6.74e4iT - 1.80e9T^{2}
73 1+1.95e4iT−2.07e9T2 1 + 1.95e4iT - 2.07e9T^{2}
79 1+4.39e4T+3.07e9T2 1 + 4.39e4T + 3.07e9T^{2}
83 1+(−7.32e4−7.32e4i)T+3.93e9iT2 1 + (-7.32e4 - 7.32e4i)T + 3.93e9iT^{2}
89 1−8.86e4iT−5.58e9T2 1 - 8.86e4iT - 5.58e9T^{2}
97 1+7.36e4T+8.58e9T2 1 + 7.36e4T + 8.58e9T^{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

−18.47743739538236651393092908756, −17.75346985063103747577406280728, −15.64505095177883040817335796060, −15.12454182340914749096485639956, −12.27938409902557999445249061825, −11.58502301029810087392385118216, −10.01022599308652725891341363539, −8.143753279263710150802104525325, −6.74994216799754702145016294433, −3.01657100555993255521868084764, 0.43939093951647305052290854050, 4.80753998200646947807092815008, 7.45027409225376373282120830400, 8.540704259673022286301772849491, 10.43719155874263528887439559439, 11.67565796960300762207303695214, 13.62289444803675834282333149306, 15.71346196556097066651134229779, 16.36340503006134644564080391006, 17.27902877545504515397432796494

Graph of the ZZ-function along the critical line