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Properties

Label 2-224-32.13-c1-0-16
Degree 22
Conductor 224224
Sign 0.963+0.268i0.963 + 0.268i
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  + (1.35 − 0.416i)2-s + (−0.244 + 0.590i)3-s + (1.65 − 1.12i)4-s + (−0.220 + 0.0913i)5-s + (−0.0844 + 0.899i)6-s + (0.707 − 0.707i)7-s + (1.76 − 2.21i)8-s + (1.83 + 1.83i)9-s + (−0.259 + 0.215i)10-s + (−0.352 − 0.851i)11-s + (0.260 + 1.25i)12-s + (−1.31 − 0.545i)13-s + (0.660 − 1.25i)14-s − 0.152i·15-s + (1.46 − 3.72i)16-s + 3.60i·17-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)2-s + (−0.141 + 0.340i)3-s + (0.826 − 0.563i)4-s + (−0.0986 + 0.0408i)5-s + (−0.0344 + 0.367i)6-s + (0.267 − 0.267i)7-s + (0.623 − 0.781i)8-s + (0.610 + 0.610i)9-s + (−0.0821 + 0.0680i)10-s + (−0.106 − 0.256i)11-s + (0.0752 + 0.361i)12-s + (−0.365 − 0.151i)13-s + (0.176 − 0.334i)14-s − 0.0393i·15-s + (0.365 − 0.930i)16-s + 0.874i·17-s + ⋯

Functional equation

Λ(s)=(224s/2 ΓC(s) L(s)=((0.963+0.268i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(224s/2 ΓC(s+1/2) L(s)=((0.963+0.268i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 224224    =    25⋅72^{5} \cdot 7
Sign: 0.963+0.268i0.963 + 0.268i
Analytic conductor: 1.788641.78864
Root analytic conductor: 1.337401.33740
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ224(141,⋅)\chi_{224} (141, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 224, ( :1/2), 0.963+0.268i)(2,\ 224,\ (\ :1/2),\ 0.963 + 0.268i)

Particular Values

L(1)L(1) ≈\approx 2.01176−0.274746i2.01176 - 0.274746i
L(12)L(\frac12) ≈\approx 2.01176−0.274746i2.01176 - 0.274746i
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+(−1.35+0.416i)T 1 + (-1.35 + 0.416i)T
7 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
good3 1+(0.244−0.590i)T+(−2.12−2.12i)T2 1 + (0.244 - 0.590i)T + (-2.12 - 2.12i)T^{2}
5 1+(0.220−0.0913i)T+(3.53−3.53i)T2 1 + (0.220 - 0.0913i)T + (3.53 - 3.53i)T^{2}
11 1+(0.352+0.851i)T+(−7.77+7.77i)T2 1 + (0.352 + 0.851i)T + (-7.77 + 7.77i)T^{2}
13 1+(1.31+0.545i)T+(9.19+9.19i)T2 1 + (1.31 + 0.545i)T + (9.19 + 9.19i)T^{2}
17 1−3.60iT−17T2 1 - 3.60iT - 17T^{2}
19 1+(3.74+1.55i)T+(13.4+13.4i)T2 1 + (3.74 + 1.55i)T + (13.4 + 13.4i)T^{2}
23 1+(0.802+0.802i)T+23iT2 1 + (0.802 + 0.802i)T + 23iT^{2}
29 1+(−0.583+1.40i)T+(−20.5−20.5i)T2 1 + (-0.583 + 1.40i)T + (-20.5 - 20.5i)T^{2}
31 1+5.77T+31T2 1 + 5.77T + 31T^{2}
37 1+(2.19−0.910i)T+(26.1−26.1i)T2 1 + (2.19 - 0.910i)T + (26.1 - 26.1i)T^{2}
41 1+(−4.11−4.11i)T+41iT2 1 + (-4.11 - 4.11i)T + 41iT^{2}
43 1+(2.07+5.01i)T+(−30.4+30.4i)T2 1 + (2.07 + 5.01i)T + (-30.4 + 30.4i)T^{2}
47 1+0.582iT−47T2 1 + 0.582iT - 47T^{2}
53 1+(−1.75−4.23i)T+(−37.4+37.4i)T2 1 + (-1.75 - 4.23i)T + (-37.4 + 37.4i)T^{2}
59 1+(7.70−3.19i)T+(41.7−41.7i)T2 1 + (7.70 - 3.19i)T + (41.7 - 41.7i)T^{2}
61 1+(−4.59+11.0i)T+(−43.1−43.1i)T2 1 + (-4.59 + 11.0i)T + (-43.1 - 43.1i)T^{2}
67 1+(3.28−7.92i)T+(−47.3−47.3i)T2 1 + (3.28 - 7.92i)T + (-47.3 - 47.3i)T^{2}
71 1+(−10.8+10.8i)T−71iT2 1 + (-10.8 + 10.8i)T - 71iT^{2}
73 1+(−9.19−9.19i)T+73iT2 1 + (-9.19 - 9.19i)T + 73iT^{2}
79 1−5.19iT−79T2 1 - 5.19iT - 79T^{2}
83 1+(−9.73−4.03i)T+(58.6+58.6i)T2 1 + (-9.73 - 4.03i)T + (58.6 + 58.6i)T^{2}
89 1+(−7.11+7.11i)T−89iT2 1 + (-7.11 + 7.11i)T - 89iT^{2}
97 1+13.2T+97T2 1 + 13.2T + 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

−12.36417639337934564793192241152, −11.12113616659095776415195394727, −10.64974138491144678528213421348, −9.650984841815102216790557545525, −8.041428058190397849579081672509, −7.00124183753048856493982514406, −5.74147636254752211161488733196, −4.67767210462556832887335887388, −3.71740997542298460032190757018, −1.99237847481692191168527346196, 2.12754267525031540590401772199, 3.80253688787973389638418188913, 4.92157451474387310511121395082, 6.12890791451371851088770287969, 7.08167428759787283357379099534, 7.963349491414491729402884347008, 9.338374010993986115330071882629, 10.62033954122241356963336672360, 11.76827995096902210031748365209, 12.33900822016598049338605674513

Graph of the ZZ-function along the critical line