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Properties

Label 2-224-7.4-c1-0-7
Degree 22
Conductor 224224
Sign −0.198+0.980i-0.198 + 0.980i
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  + (1.20 − 2.09i)3-s + (−1.91 − 3.31i)5-s + (1 + 2.44i)7-s + (−1.41 − 2.44i)9-s + (0.207 − 0.358i)11-s − 2.82·13-s − 9.24·15-s + (2.91 − 5.04i)17-s + (1.79 + 3.10i)19-s + (6.32 + 0.866i)21-s + (1.62 + 2.80i)23-s + (−4.82 + 8.36i)25-s + 0.414·27-s + 2.82·29-s + (4.20 − 7.28i)31-s + ⋯
L(s)  = 1  + (0.696 − 1.20i)3-s + (−0.856 − 1.48i)5-s + (0.377 + 0.925i)7-s + (−0.471 − 0.816i)9-s + (0.0624 − 0.108i)11-s − 0.784·13-s − 2.38·15-s + (0.706 − 1.22i)17-s + (0.411 + 0.712i)19-s + (1.38 + 0.188i)21-s + (0.338 + 0.585i)23-s + (−0.965 + 1.67i)25-s + 0.0797·27-s + 0.525·29-s + (0.755 − 1.30i)31-s + ⋯

Functional equation

Λ(s)=(224s/2 ΓC(s) L(s)=((−0.198+0.980i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(224s/2 ΓC(s+1/2) L(s)=((−0.198+0.980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 224224    =    25⋅72^{5} \cdot 7
Sign: −0.198+0.980i-0.198 + 0.980i
Analytic conductor: 1.788641.78864
Root analytic conductor: 1.337401.33740
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ224(193,⋅)\chi_{224} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 224, ( :1/2), −0.198+0.980i)(2,\ 224,\ (\ :1/2),\ -0.198 + 0.980i)

Particular Values

L(1)L(1) ≈\approx 0.844766−1.03253i0.844766 - 1.03253i
L(12)L(\frac12) ≈\approx 0.844766−1.03253i0.844766 - 1.03253i
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
7 1+(−1−2.44i)T 1 + (-1 - 2.44i)T
good3 1+(−1.20+2.09i)T+(−1.5−2.59i)T2 1 + (-1.20 + 2.09i)T + (-1.5 - 2.59i)T^{2}
5 1+(1.91+3.31i)T+(−2.5+4.33i)T2 1 + (1.91 + 3.31i)T + (-2.5 + 4.33i)T^{2}
11 1+(−0.207+0.358i)T+(−5.5−9.52i)T2 1 + (-0.207 + 0.358i)T + (-5.5 - 9.52i)T^{2}
13 1+2.82T+13T2 1 + 2.82T + 13T^{2}
17 1+(−2.91+5.04i)T+(−8.5−14.7i)T2 1 + (-2.91 + 5.04i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.79−3.10i)T+(−9.5+16.4i)T2 1 + (-1.79 - 3.10i)T + (-9.5 + 16.4i)T^{2}
23 1+(−1.62−2.80i)T+(−11.5+19.9i)T2 1 + (-1.62 - 2.80i)T + (-11.5 + 19.9i)T^{2}
29 1−2.82T+29T2 1 - 2.82T + 29T^{2}
31 1+(−4.20+7.28i)T+(−15.5−26.8i)T2 1 + (-4.20 + 7.28i)T + (-15.5 - 26.8i)T^{2}
37 1+(−1.32−2.30i)T+(−18.5+32.0i)T2 1 + (-1.32 - 2.30i)T + (-18.5 + 32.0i)T^{2}
41 1+1.17T+41T2 1 + 1.17T + 41T^{2}
43 1−1.65T+43T2 1 - 1.65T + 43T^{2}
47 1+(−3.79−6.56i)T+(−23.5+40.7i)T2 1 + (-3.79 - 6.56i)T + (-23.5 + 40.7i)T^{2}
53 1+(−0.5+0.866i)T+(−26.5−45.8i)T2 1 + (-0.5 + 0.866i)T + (-26.5 - 45.8i)T^{2}
59 1+(4.44−7.70i)T+(−29.5−51.0i)T2 1 + (4.44 - 7.70i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1.32−2.30i)T+(−30.5+52.8i)T2 1 + (-1.32 - 2.30i)T + (-30.5 + 52.8i)T^{2}
67 1+(5.62−9.73i)T+(−33.5−58.0i)T2 1 + (5.62 - 9.73i)T + (-33.5 - 58.0i)T^{2}
71 1+2.34T+71T2 1 + 2.34T + 71T^{2}
73 1+(−1.67+2.89i)T+(−36.5−63.2i)T2 1 + (-1.67 + 2.89i)T + (-36.5 - 63.2i)T^{2}
79 1+(4.03+6.98i)T+(−39.5+68.4i)T2 1 + (4.03 + 6.98i)T + (-39.5 + 68.4i)T^{2}
83 1+15.3T+83T2 1 + 15.3T + 83T^{2}
89 1+(−4.5−7.79i)T+(−44.5+77.0i)T2 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2}
97 1+6.82T+97T2 1 + 6.82T + 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

−11.98389527962239673048910508638, −11.76013992815798216651585201982, −9.617757617717251643877824008287, −8.774810531444504115688774683425, −7.942184246966596231017470418944, −7.41484753012266688206776437691, −5.67397719754822427760497833740, −4.59673674289179745639317948363, −2.77961921529794564951490862901, −1.18799524278326479230258955465, 2.91798100499612039371382808529, 3.78261020861227705606017715149, 4.73721283945033151759484524394, 6.70459373576304116152620958514, 7.57814932740326661071945422621, 8.550211721877067638018216529020, 9.991091123629378655314275699201, 10.44580801002007241902780874961, 11.18897976348401153406923243078, 12.36276636613793200641842647485

Graph of the ZZ-function along the critical line