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Properties

Label 2-224-224.109-c1-0-13
Degree 22
Conductor 224224
Sign 0.0897−0.995i0.0897 - 0.995i
Analytic cond. 1.788641.78864
Root an. cond. 1.337401.33740
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.964 + 1.03i)2-s + (0.554 + 0.722i)3-s + (−0.139 + 1.99i)4-s + (0.772 + 0.592i)5-s + (−0.212 + 1.26i)6-s + (2.49 − 0.891i)7-s + (−2.19 + 1.78i)8-s + (0.561 − 2.09i)9-s + (0.132 + 1.37i)10-s + (−3.25 − 0.428i)11-s + (−1.51 + 1.00i)12-s + (−4.64 − 1.92i)13-s + (3.32 + 1.71i)14-s + 0.886i·15-s + (−3.96 − 0.554i)16-s + (3.80 + 2.19i)17-s + ⋯
L(s)  = 1  + (0.682 + 0.731i)2-s + (0.320 + 0.417i)3-s + (−0.0695 + 0.997i)4-s + (0.345 + 0.264i)5-s + (−0.0867 + 0.518i)6-s + (0.941 − 0.337i)7-s + (−0.776 + 0.629i)8-s + (0.187 − 0.699i)9-s + (0.0417 + 0.433i)10-s + (−0.982 − 0.129i)11-s + (−0.438 + 0.290i)12-s + (−1.28 − 0.533i)13-s + (0.888 + 0.458i)14-s + 0.228i·15-s + (−0.990 − 0.138i)16-s + (0.921 + 0.532i)17-s + ⋯

Functional equation

Λ(s)=(224s/2 ΓC(s) L(s)=((0.0897−0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0897 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(224s/2 ΓC(s+1/2) L(s)=((0.0897−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0897 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 224224    =    25⋅72^{5} \cdot 7
Sign: 0.0897−0.995i0.0897 - 0.995i
Analytic conductor: 1.788641.78864
Root analytic conductor: 1.337401.33740
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ224(109,⋅)\chi_{224} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 224, ( :1/2), 0.0897−0.995i)(2,\ 224,\ (\ :1/2),\ 0.0897 - 0.995i)

Particular Values

L(1)L(1) ≈\approx 1.42990+1.30684i1.42990 + 1.30684i
L(12)L(\frac12) ≈\approx 1.42990+1.30684i1.42990 + 1.30684i
L(32)L(\frac{3}{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+(−0.964−1.03i)T 1 + (-0.964 - 1.03i)T
7 1+(−2.49+0.891i)T 1 + (-2.49 + 0.891i)T
good3 1+(−0.554−0.722i)T+(−0.776+2.89i)T2 1 + (-0.554 - 0.722i)T + (-0.776 + 2.89i)T^{2}
5 1+(−0.772−0.592i)T+(1.29+4.82i)T2 1 + (-0.772 - 0.592i)T + (1.29 + 4.82i)T^{2}
11 1+(3.25+0.428i)T+(10.6+2.84i)T2 1 + (3.25 + 0.428i)T + (10.6 + 2.84i)T^{2}
13 1+(4.64+1.92i)T+(9.19+9.19i)T2 1 + (4.64 + 1.92i)T + (9.19 + 9.19i)T^{2}
17 1+(−3.80−2.19i)T+(8.5+14.7i)T2 1 + (-3.80 - 2.19i)T + (8.5 + 14.7i)T^{2}
19 1+(−0.313−2.38i)T+(−18.3+4.91i)T2 1 + (-0.313 - 2.38i)T + (-18.3 + 4.91i)T^{2}
23 1+(1.12−4.21i)T+(−19.9−11.5i)T2 1 + (1.12 - 4.21i)T + (-19.9 - 11.5i)T^{2}
29 1+(−3.82+9.23i)T+(−20.5−20.5i)T2 1 + (-3.82 + 9.23i)T + (-20.5 - 20.5i)T^{2}
31 1+(−0.642+1.11i)T+(−15.5−26.8i)T2 1 + (-0.642 + 1.11i)T + (-15.5 - 26.8i)T^{2}
37 1+(3.26+2.50i)T+(9.57+35.7i)T2 1 + (3.26 + 2.50i)T + (9.57 + 35.7i)T^{2}
41 1+(−4.76−4.76i)T+41iT2 1 + (-4.76 - 4.76i)T + 41iT^{2}
43 1+(0.0816+0.197i)T+(−30.4+30.4i)T2 1 + (0.0816 + 0.197i)T + (-30.4 + 30.4i)T^{2}
47 1+(5.27−3.04i)T+(23.5−40.7i)T2 1 + (5.27 - 3.04i)T + (23.5 - 40.7i)T^{2}
53 1+(6.41+0.844i)T+(51.1+13.7i)T2 1 + (6.41 + 0.844i)T + (51.1 + 13.7i)T^{2}
59 1+(1.84−14.0i)T+(−56.9−15.2i)T2 1 + (1.84 - 14.0i)T + (-56.9 - 15.2i)T^{2}
61 1+(7.84−1.03i)T+(58.9−15.7i)T2 1 + (7.84 - 1.03i)T + (58.9 - 15.7i)T^{2}
67 1+(−4.59−5.98i)T+(−17.3+64.7i)T2 1 + (-4.59 - 5.98i)T + (-17.3 + 64.7i)T^{2}
71 1+(−6.86+6.86i)T−71iT2 1 + (-6.86 + 6.86i)T - 71iT^{2}
73 1+(−9.46+2.53i)T+(63.2−36.5i)T2 1 + (-9.46 + 2.53i)T + (63.2 - 36.5i)T^{2}
79 1+(6.16−3.56i)T+(39.5−68.4i)T2 1 + (6.16 - 3.56i)T + (39.5 - 68.4i)T^{2}
83 1+(−4.32−1.79i)T+(58.6+58.6i)T2 1 + (-4.32 - 1.79i)T + (58.6 + 58.6i)T^{2}
89 1+(−7.96−2.13i)T+(77.0+44.5i)T2 1 + (-7.96 - 2.13i)T + (77.0 + 44.5i)T^{2}
97 1+16.7T+97T2 1 + 16.7T + 97T^{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

−12.54643043269646391687899717513, −11.81651557861617110453756797960, −10.40524994316988896963859189911, −9.619590857924990482167219780943, −8.051633665787009072435760190793, −7.68861336112144036318647279642, −6.18338242885071771634721666875, −5.15976750495678281683320915056, −4.06203341857613610593257723879, −2.70078029338222087305293342878, 1.78173591637930089624515762803, 2.79771041684807355233950965677, 4.97611392191706454717491509097, 5.09110468456966997909273614705, 6.96943215059874585649579564281, 8.038128273125078206738921070979, 9.266541954939489839265925162475, 10.29726462020507996147510454083, 11.14939794454463873838733542944, 12.30689345380477136546485324536

Graph of the ZZ-function along the critical line