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Properties

Label 2-28-28.3-c5-0-7
Degree 22
Conductor 2828
Sign −0.469−0.882i-0.469 - 0.882i
Analytic cond. 4.490744.49074
Root an. cond. 2.119132.11913
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  + (4.87 + 2.86i)2-s + (−5.07 + 8.78i)3-s + (15.5 + 27.9i)4-s + (−73.9 + 42.6i)5-s + (−49.9 + 28.2i)6-s + (80.3 − 101. i)7-s + (−4.39 + 180. i)8-s + (70.0 + 121. i)9-s + (−482. − 3.91i)10-s + (256. + 148. i)11-s + (−324. − 5.26i)12-s − 324. i·13-s + (683. − 265. i)14-s − 866. i·15-s + (−540. + 869. i)16-s + (1.37e3 + 795. i)17-s + ⋯
L(s)  = 1  + (0.861 + 0.506i)2-s + (−0.325 + 0.563i)3-s + (0.485 + 0.874i)4-s + (−1.32 + 0.763i)5-s + (−0.566 + 0.320i)6-s + (0.620 − 0.784i)7-s + (−0.0242 + 0.999i)8-s + (0.288 + 0.498i)9-s + (−1.52 − 0.0123i)10-s + (0.640 + 0.369i)11-s + (−0.650 − 0.0105i)12-s − 0.533i·13-s + (0.932 − 0.361i)14-s − 0.994i·15-s + (−0.527 + 0.849i)16-s + (1.15 + 0.667i)17-s + ⋯

Functional equation

Λ(s)=(28s/2 ΓC(s) L(s)=((−0.469−0.882i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(28s/2 ΓC(s+5/2) L(s)=((−0.469−0.882i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2828    =    22⋅72^{2} \cdot 7
Sign: −0.469−0.882i-0.469 - 0.882i
Analytic conductor: 4.490744.49074
Root analytic conductor: 2.119132.11913
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ28(3,⋅)\chi_{28} (3, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 28, ( :5/2), −0.469−0.882i)(2,\ 28,\ (\ :5/2),\ -0.469 - 0.882i)

Particular Values

L(3)L(3) ≈\approx 0.969562+1.61393i0.969562 + 1.61393i
L(12)L(\frac12) ≈\approx 0.969562+1.61393i0.969562 + 1.61393i
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+(−4.87−2.86i)T 1 + (-4.87 - 2.86i)T
7 1+(−80.3+101.i)T 1 + (-80.3 + 101. i)T
good3 1+(5.07−8.78i)T+(−121.5−210.i)T2 1 + (5.07 - 8.78i)T + (-121.5 - 210. i)T^{2}
5 1+(73.9−42.6i)T+(1.56e3−2.70e3i)T2 1 + (73.9 - 42.6i)T + (1.56e3 - 2.70e3i)T^{2}
11 1+(−256.−148.i)T+(8.05e4+1.39e5i)T2 1 + (-256. - 148. i)T + (8.05e4 + 1.39e5i)T^{2}
13 1+324.iT−3.71e5T2 1 + 324. iT - 3.71e5T^{2}
17 1+(−1.37e3−795.i)T+(7.09e5+1.22e6i)T2 1 + (-1.37e3 - 795. i)T + (7.09e5 + 1.22e6i)T^{2}
19 1+(587.+1.01e3i)T+(−1.23e6+2.14e6i)T2 1 + (587. + 1.01e3i)T + (-1.23e6 + 2.14e6i)T^{2}
23 1+(−4.09e3+2.36e3i)T+(3.21e6−5.57e6i)T2 1 + (-4.09e3 + 2.36e3i)T + (3.21e6 - 5.57e6i)T^{2}
29 1+4.56e3T+2.05e7T2 1 + 4.56e3T + 2.05e7T^{2}
31 1+(987.−1.71e3i)T+(−1.43e7−2.47e7i)T2 1 + (987. - 1.71e3i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1+(−669.−1.15e3i)T+(−3.46e7+6.00e7i)T2 1 + (-669. - 1.15e3i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1−9.20e3iT−1.15e8T2 1 - 9.20e3iT - 1.15e8T^{2}
43 1−1.07e3iT−1.47e8T2 1 - 1.07e3iT - 1.47e8T^{2}
47 1+(−1.11e3−1.92e3i)T+(−1.14e8+1.98e8i)T2 1 + (-1.11e3 - 1.92e3i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(−4.12e3+7.14e3i)T+(−2.09e8−3.62e8i)T2 1 + (-4.12e3 + 7.14e3i)T + (-2.09e8 - 3.62e8i)T^{2}
59 1+(−2.06e4+3.58e4i)T+(−3.57e8−6.19e8i)T2 1 + (-2.06e4 + 3.58e4i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(−1.17e4+6.79e3i)T+(4.22e8−7.31e8i)T2 1 + (-1.17e4 + 6.79e3i)T + (4.22e8 - 7.31e8i)T^{2}
67 1+(−2.42e4−1.40e4i)T+(6.75e8+1.16e9i)T2 1 + (-2.42e4 - 1.40e4i)T + (6.75e8 + 1.16e9i)T^{2}
71 1+6.32e4iT−1.80e9T2 1 + 6.32e4iT - 1.80e9T^{2}
73 1+(1.16e4+6.71e3i)T+(1.03e9+1.79e9i)T2 1 + (1.16e4 + 6.71e3i)T + (1.03e9 + 1.79e9i)T^{2}
79 1+(9.71e3−5.61e3i)T+(1.53e9−2.66e9i)T2 1 + (9.71e3 - 5.61e3i)T + (1.53e9 - 2.66e9i)T^{2}
83 1+3.04e4T+3.93e9T2 1 + 3.04e4T + 3.93e9T^{2}
89 1+(−6.56e4+3.79e4i)T+(2.79e9−4.83e9i)T2 1 + (-6.56e4 + 3.79e4i)T + (2.79e9 - 4.83e9i)T^{2}
97 1−7.93e4iT−8.58e9T2 1 - 7.93e4iT - 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

−16.38983757362883077741397255025, −15.10869344991163992642141606196, −14.60918905746780055675025486603, −12.89246339073494809753060311571, −11.42325157483841965293710568919, −10.67229411491704566302210808152, −7.983762009783164362111964756124, −7.00214812438890266848349908494, −4.81125269373498512964341485557, −3.64124437612353444007345871687, 1.13586621627236100520069221193, 3.84643567060097433571843150918, 5.47579834116459124969990319605, 7.32345030565468155991835038390, 9.125704616579816536461780684323, 11.47910971606163952155102642835, 11.91130761546460204632821135714, 12.87246623034947686088743895832, 14.56520940099580017509565146501, 15.52314343968041215886710187322

Graph of the ZZ-function along the critical line