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Properties

Label 2-13-13.10-c9-0-5
Degree 22
Conductor 1313
Sign 0.498−0.867i0.498 - 0.867i
Analytic cond. 6.695466.69546
Root an. cond. 2.587552.58755
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (17.5 − 10.1i)2-s + (49.6 + 86.0i)3-s + (−50.8 + 88.0i)4-s + 1.79e3i·5-s + (1.74e3 + 1.00e3i)6-s + (−1.67e3 − 967. i)7-s + 1.24e4i·8-s + (4.91e3 − 8.50e3i)9-s + (1.82e4 + 3.15e4i)10-s + (4.64e4 − 2.68e4i)11-s − 1.00e4·12-s + (1.53e3 + 1.02e5i)13-s − 3.92e4·14-s + (−1.54e5 + 8.93e4i)15-s + (9.98e4 + 1.72e5i)16-s + (1.98e5 − 3.43e5i)17-s + ⋯
L(s)  = 1  + (0.775 − 0.447i)2-s + (0.353 + 0.613i)3-s + (−0.0992 + 0.171i)4-s + 1.28i·5-s + (0.548 + 0.316i)6-s + (−0.263 − 0.152i)7-s + 1.07i·8-s + (0.249 − 0.432i)9-s + (0.576 + 0.998i)10-s + (0.957 − 0.552i)11-s − 0.140·12-s + (0.0149 + 0.999i)13-s − 0.272·14-s + (−0.789 + 0.455i)15-s + (0.380 + 0.659i)16-s + (0.575 − 0.996i)17-s + ⋯

Functional equation

Λ(s)=(13s/2 ΓC(s) L(s)=((0.498−0.867i) Λ‾(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 - 0.867i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(13s/2 ΓC(s+9/2) L(s)=((0.498−0.867i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.498 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1313
Sign: 0.498−0.867i0.498 - 0.867i
Analytic conductor: 6.695466.69546
Root analytic conductor: 2.587552.58755
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: χ13(10,⋅)\chi_{13} (10, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 13, ( :9/2), 0.498−0.867i)(2,\ 13,\ (\ :9/2),\ 0.498 - 0.867i)

Particular Values

L(5)L(5) ≈\approx 2.13576+1.23610i2.13576 + 1.23610i
L(12)L(\frac12) ≈\approx 2.13576+1.23610i2.13576 + 1.23610i
L(112)L(\frac{11}{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)
bad13 1+(−1.53e3−1.02e5i)T 1 + (-1.53e3 - 1.02e5i)T
good2 1+(−17.5+10.1i)T+(256−443.i)T2 1 + (-17.5 + 10.1i)T + (256 - 443. i)T^{2}
3 1+(−49.6−86.0i)T+(−9.84e3+1.70e4i)T2 1 + (-49.6 - 86.0i)T + (-9.84e3 + 1.70e4i)T^{2}
5 1−1.79e3iT−1.95e6T2 1 - 1.79e3iT - 1.95e6T^{2}
7 1+(1.67e3+967.i)T+(2.01e7+3.49e7i)T2 1 + (1.67e3 + 967. i)T + (2.01e7 + 3.49e7i)T^{2}
11 1+(−4.64e4+2.68e4i)T+(1.17e9−2.04e9i)T2 1 + (-4.64e4 + 2.68e4i)T + (1.17e9 - 2.04e9i)T^{2}
17 1+(−1.98e5+3.43e5i)T+(−5.92e10−1.02e11i)T2 1 + (-1.98e5 + 3.43e5i)T + (-5.92e10 - 1.02e11i)T^{2}
19 1+(4.35e5+2.51e5i)T+(1.61e11+2.79e11i)T2 1 + (4.35e5 + 2.51e5i)T + (1.61e11 + 2.79e11i)T^{2}
23 1+(5.98e5+1.03e6i)T+(−9.00e11+1.55e12i)T2 1 + (5.98e5 + 1.03e6i)T + (-9.00e11 + 1.55e12i)T^{2}
29 1+(−2.44e6−4.23e6i)T+(−7.25e12+1.25e13i)T2 1 + (-2.44e6 - 4.23e6i)T + (-7.25e12 + 1.25e13i)T^{2}
31 1−2.72e6iT−2.64e13T2 1 - 2.72e6iT - 2.64e13T^{2}
37 1+(−5.29e6+3.05e6i)T+(6.49e13−1.12e14i)T2 1 + (-5.29e6 + 3.05e6i)T + (6.49e13 - 1.12e14i)T^{2}
41 1+(−2.28e7+1.31e7i)T+(1.63e14−2.83e14i)T2 1 + (-2.28e7 + 1.31e7i)T + (1.63e14 - 2.83e14i)T^{2}
43 1+(−8.62e6+1.49e7i)T+(−2.51e14−4.35e14i)T2 1 + (-8.62e6 + 1.49e7i)T + (-2.51e14 - 4.35e14i)T^{2}
47 1−1.60e7iT−1.11e15T2 1 - 1.60e7iT - 1.11e15T^{2}
53 1+4.91e7T+3.29e15T2 1 + 4.91e7T + 3.29e15T^{2}
59 1+(−4.20e7−2.42e7i)T+(4.33e15+7.50e15i)T2 1 + (-4.20e7 - 2.42e7i)T + (4.33e15 + 7.50e15i)T^{2}
61 1+(6.58e7−1.13e8i)T+(−5.84e15−1.01e16i)T2 1 + (6.58e7 - 1.13e8i)T + (-5.84e15 - 1.01e16i)T^{2}
67 1+(−1.08e8+6.27e7i)T+(1.36e16−2.35e16i)T2 1 + (-1.08e8 + 6.27e7i)T + (1.36e16 - 2.35e16i)T^{2}
71 1+(2.59e8+1.50e8i)T+(2.29e16+3.97e16i)T2 1 + (2.59e8 + 1.50e8i)T + (2.29e16 + 3.97e16i)T^{2}
73 1+2.58e8iT−5.88e16T2 1 + 2.58e8iT - 5.88e16T^{2}
79 1−8.39e7T+1.19e17T2 1 - 8.39e7T + 1.19e17T^{2}
83 1−3.52e8iT−1.86e17T2 1 - 3.52e8iT - 1.86e17T^{2}
89 1+(9.87e8−5.70e8i)T+(1.75e17−3.03e17i)T2 1 + (9.87e8 - 5.70e8i)T + (1.75e17 - 3.03e17i)T^{2}
97 1+(−1.20e9−6.95e8i)T+(3.80e17+6.58e17i)T2 1 + (-1.20e9 - 6.95e8i)T + (3.80e17 + 6.58e17i)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

−18.06024108737122654753627674293, −16.36374141568508519001833085818, −14.59541987219427089310110095279, −14.06104264058391999902159098262, −12.15509077812315234236770069335, −10.81380272556161080379640685687, −9.089341505560391481275122465010, −6.71752363264388211156255735227, −4.13879548444323766298831068957, −2.94707842734536605108552284126, 1.24618016669424069828012787539, 4.35872015339458809919735330603, 6.01243077329158026047139753416, 8.028917077045440777336303530146, 9.771396616052600588332256376497, 12.52736044967245567091511105964, 13.09670915388072914942712160636, 14.53296616471286558187931271811, 15.88376795600529753958998267596, 17.27005423546639201265533939282

Graph of the ZZ-function along the critical line