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Properties

Label 2-224-32.29-c1-0-3
Degree 22
Conductor 224224
Sign 0.527+0.849i0.527 + 0.849i
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  + (−1.39 − 0.230i)2-s + (−3.03 − 1.25i)3-s + (1.89 + 0.643i)4-s + (0.551 + 1.33i)5-s + (3.94 + 2.45i)6-s + (−0.707 + 0.707i)7-s + (−2.49 − 1.33i)8-s + (5.50 + 5.50i)9-s + (−0.462 − 1.98i)10-s + (−0.508 + 0.210i)11-s + (−4.93 − 4.33i)12-s + (1.29 − 3.13i)13-s + (1.14 − 0.823i)14-s − 4.73i·15-s + (3.17 + 2.43i)16-s − 3.89i·17-s + ⋯
L(s)  = 1  + (−0.986 − 0.163i)2-s + (−1.75 − 0.725i)3-s + (0.946 + 0.321i)4-s + (0.246 + 0.595i)5-s + (1.61 + 1.00i)6-s + (−0.267 + 0.267i)7-s + (−0.881 − 0.471i)8-s + (1.83 + 1.83i)9-s + (−0.146 − 0.627i)10-s + (−0.153 + 0.0635i)11-s + (−1.42 − 1.25i)12-s + (0.360 − 0.869i)13-s + (0.307 − 0.220i)14-s − 1.22i·15-s + (0.793 + 0.609i)16-s − 0.944i·17-s + ⋯

Functional equation

Λ(s)=(224s/2 ΓC(s) L(s)=((0.527+0.849i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(224s/2 ΓC(s+1/2) L(s)=((0.527+0.849i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 224224    =    25⋅72^{5} \cdot 7
Sign: 0.527+0.849i0.527 + 0.849i
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), 0.527+0.849i)(2,\ 224,\ (\ :1/2),\ 0.527 + 0.849i)

Particular Values

L(1)L(1) ≈\approx 0.390258−0.217038i0.390258 - 0.217038i
L(12)L(\frac12) ≈\approx 0.390258−0.217038i0.390258 - 0.217038i
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+(1.39+0.230i)T 1 + (1.39 + 0.230i)T
7 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
good3 1+(3.03+1.25i)T+(2.12+2.12i)T2 1 + (3.03 + 1.25i)T + (2.12 + 2.12i)T^{2}
5 1+(−0.551−1.33i)T+(−3.53+3.53i)T2 1 + (-0.551 - 1.33i)T + (-3.53 + 3.53i)T^{2}
11 1+(0.508−0.210i)T+(7.77−7.77i)T2 1 + (0.508 - 0.210i)T + (7.77 - 7.77i)T^{2}
13 1+(−1.29+3.13i)T+(−9.19−9.19i)T2 1 + (-1.29 + 3.13i)T + (-9.19 - 9.19i)T^{2}
17 1+3.89iT−17T2 1 + 3.89iT - 17T^{2}
19 1+(−2.18+5.28i)T+(−13.4−13.4i)T2 1 + (-2.18 + 5.28i)T + (-13.4 - 13.4i)T^{2}
23 1+(−3.28−3.28i)T+23iT2 1 + (-3.28 - 3.28i)T + 23iT^{2}
29 1+(5.84+2.42i)T+(20.5+20.5i)T2 1 + (5.84 + 2.42i)T + (20.5 + 20.5i)T^{2}
31 1−7.47T+31T2 1 - 7.47T + 31T^{2}
37 1+(−2.00−4.83i)T+(−26.1+26.1i)T2 1 + (-2.00 - 4.83i)T + (-26.1 + 26.1i)T^{2}
41 1+(−2.71−2.71i)T+41iT2 1 + (-2.71 - 2.71i)T + 41iT^{2}
43 1+(−5.33+2.20i)T+(30.4−30.4i)T2 1 + (-5.33 + 2.20i)T + (30.4 - 30.4i)T^{2}
47 1−3.49iT−47T2 1 - 3.49iT - 47T^{2}
53 1+(−8.42+3.49i)T+(37.4−37.4i)T2 1 + (-8.42 + 3.49i)T + (37.4 - 37.4i)T^{2}
59 1+(3.20+7.73i)T+(−41.7+41.7i)T2 1 + (3.20 + 7.73i)T + (-41.7 + 41.7i)T^{2}
61 1+(1.33+0.554i)T+(43.1+43.1i)T2 1 + (1.33 + 0.554i)T + (43.1 + 43.1i)T^{2}
67 1+(−9.53−3.94i)T+(47.3+47.3i)T2 1 + (-9.53 - 3.94i)T + (47.3 + 47.3i)T^{2}
71 1+(−7.97+7.97i)T−71iT2 1 + (-7.97 + 7.97i)T - 71iT^{2}
73 1+(9.49+9.49i)T+73iT2 1 + (9.49 + 9.49i)T + 73iT^{2}
79 1+5.36iT−79T2 1 + 5.36iT - 79T^{2}
83 1+(6.45−15.5i)T+(−58.6−58.6i)T2 1 + (6.45 - 15.5i)T + (-58.6 - 58.6i)T^{2}
89 1+(0.200−0.200i)T−89iT2 1 + (0.200 - 0.200i)T - 89iT^{2}
97 1−6.47T+97T2 1 - 6.47T + 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

−11.70673046331558625199185015157, −11.19045503203091727308113355759, −10.39431644089131241232230481127, −9.455458889718633677983208421918, −7.81631433379077509576809174276, −6.97291898580437526752683307398, −6.22274619818210272138754310759, −5.18022829710251644046078302120, −2.67885288151935544985029575324, −0.78748888238355980547171462934, 1.16016815682476189996781967171, 4.05877797658601182146429201318, 5.45786679750590722034045566163, 6.17479855752869090142560525458, 7.19544419147975215436335534599, 8.783237582730334187977848072033, 9.688695915651262297570180749778, 10.48934705113844988999996549520, 11.15936594451774573682524109396, 12.09320098047866831267470500661

Graph of the ZZ-function along the critical line