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Properties

Label 2-224-32.29-c1-0-9
Degree 22
Conductor 224224
Sign 0.967+0.254i0.967 + 0.254i
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  + (−0.420 − 1.35i)2-s + (2.28 + 0.946i)3-s + (−1.64 + 1.13i)4-s + (0.446 + 1.07i)5-s + (0.316 − 3.48i)6-s + (−0.707 + 0.707i)7-s + (2.22 + 1.74i)8-s + (2.20 + 2.20i)9-s + (1.26 − 1.05i)10-s + (2.89 − 1.19i)11-s + (−4.83 + 1.03i)12-s + (1.10 − 2.65i)13-s + (1.25 + 0.657i)14-s + 2.88i·15-s + (1.41 − 3.73i)16-s + 3.12i·17-s + ⋯
L(s)  = 1  + (−0.297 − 0.954i)2-s + (1.31 + 0.546i)3-s + (−0.823 + 0.567i)4-s + (0.199 + 0.481i)5-s + (0.129 − 1.42i)6-s + (−0.267 + 0.267i)7-s + (0.786 + 0.616i)8-s + (0.734 + 0.734i)9-s + (0.400 − 0.333i)10-s + (0.871 − 0.361i)11-s + (−1.39 + 0.299i)12-s + (0.305 − 0.737i)13-s + (0.334 + 0.175i)14-s + 0.744i·15-s + (0.354 − 0.934i)16-s + 0.758i·17-s + ⋯

Functional equation

Λ(s)=(224s/2 ΓC(s) L(s)=((0.967+0.254i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(224s/2 ΓC(s+1/2) L(s)=((0.967+0.254i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 224224    =    25⋅72^{5} \cdot 7
Sign: 0.967+0.254i0.967 + 0.254i
Analytic conductor: 1.788641.78864
Root analytic conductor: 1.337401.33740
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ224(29,⋅)\chi_{224} (29, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 224, ( :1/2), 0.967+0.254i)(2,\ 224,\ (\ :1/2),\ 0.967 + 0.254i)

Particular Values

L(1)L(1) ≈\approx 1.48180−0.191665i1.48180 - 0.191665i
L(12)L(\frac12) ≈\approx 1.48180−0.191665i1.48180 - 0.191665i
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+(0.420+1.35i)T 1 + (0.420 + 1.35i)T
7 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
good3 1+(−2.28−0.946i)T+(2.12+2.12i)T2 1 + (-2.28 - 0.946i)T + (2.12 + 2.12i)T^{2}
5 1+(−0.446−1.07i)T+(−3.53+3.53i)T2 1 + (-0.446 - 1.07i)T + (-3.53 + 3.53i)T^{2}
11 1+(−2.89+1.19i)T+(7.77−7.77i)T2 1 + (-2.89 + 1.19i)T + (7.77 - 7.77i)T^{2}
13 1+(−1.10+2.65i)T+(−9.19−9.19i)T2 1 + (-1.10 + 2.65i)T + (-9.19 - 9.19i)T^{2}
17 1−3.12iT−17T2 1 - 3.12iT - 17T^{2}
19 1+(1.63−3.95i)T+(−13.4−13.4i)T2 1 + (1.63 - 3.95i)T + (-13.4 - 13.4i)T^{2}
23 1+(1.37+1.37i)T+23iT2 1 + (1.37 + 1.37i)T + 23iT^{2}
29 1+(8.15+3.37i)T+(20.5+20.5i)T2 1 + (8.15 + 3.37i)T + (20.5 + 20.5i)T^{2}
31 1+2.07T+31T2 1 + 2.07T + 31T^{2}
37 1+(2.47+5.96i)T+(−26.1+26.1i)T2 1 + (2.47 + 5.96i)T + (-26.1 + 26.1i)T^{2}
41 1+(5.68+5.68i)T+41iT2 1 + (5.68 + 5.68i)T + 41iT^{2}
43 1+(−5.63+2.33i)T+(30.4−30.4i)T2 1 + (-5.63 + 2.33i)T + (30.4 - 30.4i)T^{2}
47 1−8.41iT−47T2 1 - 8.41iT - 47T^{2}
53 1+(12.4−5.16i)T+(37.4−37.4i)T2 1 + (12.4 - 5.16i)T + (37.4 - 37.4i)T^{2}
59 1+(−2.20−5.32i)T+(−41.7+41.7i)T2 1 + (-2.20 - 5.32i)T + (-41.7 + 41.7i)T^{2}
61 1+(−4.89−2.02i)T+(43.1+43.1i)T2 1 + (-4.89 - 2.02i)T + (43.1 + 43.1i)T^{2}
67 1+(−8.29−3.43i)T+(47.3+47.3i)T2 1 + (-8.29 - 3.43i)T + (47.3 + 47.3i)T^{2}
71 1+(−6.35+6.35i)T−71iT2 1 + (-6.35 + 6.35i)T - 71iT^{2}
73 1+(10.6+10.6i)T+73iT2 1 + (10.6 + 10.6i)T + 73iT^{2}
79 1−5.81iT−79T2 1 - 5.81iT - 79T^{2}
83 1+(1.91−4.62i)T+(−58.6−58.6i)T2 1 + (1.91 - 4.62i)T + (-58.6 - 58.6i)T^{2}
89 1+(−5.79+5.79i)T−89iT2 1 + (-5.79 + 5.79i)T - 89iT^{2}
97 1−13.8T+97T2 1 - 13.8T + 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

−12.27137396885066904687880781729, −10.94557748636939011998116246031, −10.22830564268049249689419073921, −9.292013894446044270536117356362, −8.628657093929688557906532704846, −7.71911397627063791466536126251, −5.95373136539693982063021914379, −4.02766970452384769424172380369, −3.35250040497967046038613686707, −2.09967920408832612637111031638, 1.63372582738785931005600295559, 3.64169716848360266283034707439, 4.99475732920135165807201801437, 6.64562693086589951070332328279, 7.24485934754130908292217007268, 8.402341960555268608076160067882, 9.164878183352105001720478028976, 9.640268155984212018461125683629, 11.32471952101022801504753017670, 12.88709313463125116171581893685

Graph of the ZZ-function along the critical line