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Properties

Label 2-21-21.17-c17-0-11
Degree 22
Conductor 2121
Sign 0.0926+0.995i0.0926 + 0.995i
Analytic cond. 38.476638.4766
Root an. cond. 6.202956.20295
Motivic weight 1717
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−389. + 224. i)2-s + (−6.71e3 + 9.16e3i)3-s + (3.54e4 − 6.13e4i)4-s + (6.52e5 + 1.12e6i)5-s + (5.54e5 − 5.07e6i)6-s + (1.13e6 + 1.52e7i)7-s − 2.70e7i·8-s + (−3.89e7 − 1.23e8i)9-s + (−5.07e8 − 2.92e8i)10-s + (7.67e8 + 4.43e8i)11-s + (3.24e8 + 7.36e8i)12-s + 1.92e9i·13-s + (−3.86e9 − 5.66e9i)14-s + (−1.47e10 − 1.60e9i)15-s + (1.07e10 + 1.85e10i)16-s + (−1.05e10 + 1.83e10i)17-s + ⋯
L(s)  = 1  + (−1.07 + 0.620i)2-s + (−0.591 + 0.806i)3-s + (0.270 − 0.467i)4-s + (0.746 + 1.29i)5-s + (0.134 − 1.23i)6-s + (0.0747 + 0.997i)7-s − 0.570i·8-s + (−0.301 − 0.953i)9-s + (−1.60 − 0.926i)10-s + (1.07 + 0.623i)11-s + (0.217 + 0.494i)12-s + 0.653i·13-s + (−0.699 − 1.02i)14-s + (−1.48 − 0.162i)15-s + (0.624 + 1.08i)16-s + (−0.367 + 0.637i)17-s + ⋯

Functional equation

Λ(s)=(21s/2 ΓC(s) L(s)=((0.0926+0.995i) Λ‾(18−s)\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0926 + 0.995i)\, \overline{\Lambda}(18-s) \end{aligned}
Λ(s)=(21s/2 ΓC(s+17/2) L(s)=((0.0926+0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.0926 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2121    =    3⋅73 \cdot 7
Sign: 0.0926+0.995i0.0926 + 0.995i
Analytic conductor: 38.476638.4766
Root analytic conductor: 6.202956.20295
Motivic weight: 1717
Rational: no
Arithmetic: yes
Character: χ21(17,⋅)\chi_{21} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 21, ( :17/2), 0.0926+0.995i)(2,\ 21,\ (\ :17/2),\ 0.0926 + 0.995i)

Particular Values

L(9)L(9) ≈\approx 0.75954430950.7595443095
L(12)L(\frac12) ≈\approx 0.75954430950.7595443095
L(192)L(\frac{19}{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)
bad3 1+(6.71e3−9.16e3i)T 1 + (6.71e3 - 9.16e3i)T
7 1+(−1.13e6−1.52e7i)T 1 + (-1.13e6 - 1.52e7i)T
good2 1+(389.−224.i)T+(6.55e4−1.13e5i)T2 1 + (389. - 224. i)T + (6.55e4 - 1.13e5i)T^{2}
5 1+(−6.52e5−1.12e6i)T+(−3.81e11+6.60e11i)T2 1 + (-6.52e5 - 1.12e6i)T + (-3.81e11 + 6.60e11i)T^{2}
11 1+(−7.67e8−4.43e8i)T+(2.52e17+4.37e17i)T2 1 + (-7.67e8 - 4.43e8i)T + (2.52e17 + 4.37e17i)T^{2}
13 1−1.92e9iT−8.65e18T2 1 - 1.92e9iT - 8.65e18T^{2}
17 1+(1.05e10−1.83e10i)T+(−4.13e20−7.16e20i)T2 1 + (1.05e10 - 1.83e10i)T + (-4.13e20 - 7.16e20i)T^{2}
19 1+(1.02e11−5.93e10i)T+(2.74e21−4.74e21i)T2 1 + (1.02e11 - 5.93e10i)T + (2.74e21 - 4.74e21i)T^{2}
23 1+(2.29e11−1.32e11i)T+(7.05e22−1.22e23i)T2 1 + (2.29e11 - 1.32e11i)T + (7.05e22 - 1.22e23i)T^{2}
29 1−2.01e12iT−7.25e24T2 1 - 2.01e12iT - 7.25e24T^{2}
31 1+(−1.38e12−7.99e11i)T+(1.12e25+1.95e25i)T2 1 + (-1.38e12 - 7.99e11i)T + (1.12e25 + 1.95e25i)T^{2}
37 1+(6.92e12+1.19e13i)T+(−2.28e26+3.95e26i)T2 1 + (6.92e12 + 1.19e13i)T + (-2.28e26 + 3.95e26i)T^{2}
41 1−4.36e13T+2.61e27T2 1 - 4.36e13T + 2.61e27T^{2}
43 1+4.58e12T+5.87e27T2 1 + 4.58e12T + 5.87e27T^{2}
47 1+(9.87e13+1.71e14i)T+(−1.33e28+2.30e28i)T2 1 + (9.87e13 + 1.71e14i)T + (-1.33e28 + 2.30e28i)T^{2}
53 1+(−6.00e14−3.46e14i)T+(1.02e29+1.77e29i)T2 1 + (-6.00e14 - 3.46e14i)T + (1.02e29 + 1.77e29i)T^{2}
59 1+(−1.85e14+3.21e14i)T+(−6.35e29−1.10e30i)T2 1 + (-1.85e14 + 3.21e14i)T + (-6.35e29 - 1.10e30i)T^{2}
61 1+(1.54e15−8.90e14i)T+(1.12e30−1.94e30i)T2 1 + (1.54e15 - 8.90e14i)T + (1.12e30 - 1.94e30i)T^{2}
67 1+(−1.02e15+1.76e15i)T+(−5.52e30−9.56e30i)T2 1 + (-1.02e15 + 1.76e15i)T + (-5.52e30 - 9.56e30i)T^{2}
71 1−5.60e15iT−2.96e31T2 1 - 5.60e15iT - 2.96e31T^{2}
73 1+(−9.29e15−5.36e15i)T+(2.37e31+4.11e31i)T2 1 + (-9.29e15 - 5.36e15i)T + (2.37e31 + 4.11e31i)T^{2}
79 1+(6.38e15+1.10e16i)T+(−9.09e31+1.57e32i)T2 1 + (6.38e15 + 1.10e16i)T + (-9.09e31 + 1.57e32i)T^{2}
83 1−3.82e16T+4.21e32T2 1 - 3.82e16T + 4.21e32T^{2}
89 1+(−3.67e16−6.36e16i)T+(−6.89e32+1.19e33i)T2 1 + (-3.67e16 - 6.36e16i)T + (-6.89e32 + 1.19e33i)T^{2}
97 1+7.45e16iT−5.95e33T2 1 + 7.45e16iT - 5.95e33T^{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

−15.23506577871801557740116876778, −14.54482729743783137295114299655, −12.23365653196114838894636060719, −10.72810320885232847581357439720, −9.733590436513689645721901915550, −8.778514881812824285829852964582, −6.73250245850815798639312167042, −6.05773479606398236327301315424, −3.89100820202607032290236532250, −1.90434340722351658857077963479, 0.44494927004903053753905693619, 0.921022297231434398771283467317, 2.07072471702264489894218209926, 4.76214595221735092385552694121, 6.28489773819051555586006365743, 8.073669726664099352462172121805, 9.146972415605823820333998437107, 10.50354905779037299336444019863, 11.64337748386174819816825712207, 13.00878609223045732529946606613

Graph of the ZZ-function along the critical line