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Properties

Label 2-22e2-11.3-c1-0-2
Degree 22
Conductor 484484
Sign 0.266−0.963i0.266 - 0.963i
Analytic cond. 3.864753.86475
Root an. cond. 1.965891.96589
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  + (0.309 + 0.951i)3-s + (2.42 + 1.76i)5-s + (−0.618 + 1.90i)7-s + (1.61 − 1.17i)9-s + (−3.23 + 2.35i)13-s + (−0.927 + 2.85i)15-s + (4.85 + 3.52i)17-s + (−2.47 − 7.60i)19-s − 1.99·21-s − 3·23-s + (1.23 + 3.80i)25-s + (4.04 + 2.93i)27-s + (−4.04 + 2.93i)31-s + (−4.85 + 3.52i)35-s + (−0.309 + 0.951i)37-s + ⋯
L(s)  = 1  + (0.178 + 0.549i)3-s + (1.08 + 0.788i)5-s + (−0.233 + 0.718i)7-s + (0.539 − 0.391i)9-s + (−0.897 + 0.652i)13-s + (−0.239 + 0.736i)15-s + (1.17 + 0.855i)17-s + (−0.567 − 1.74i)19-s − 0.436·21-s − 0.625·23-s + (0.247 + 0.760i)25-s + (0.778 + 0.565i)27-s + (−0.726 + 0.527i)31-s + (−0.820 + 0.596i)35-s + (−0.0508 + 0.156i)37-s + ⋯

Functional equation

Λ(s)=(484s/2 ΓC(s) L(s)=((0.266−0.963i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(484s/2 ΓC(s+1/2) L(s)=((0.266−0.963i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 484484    =    22⋅1122^{2} \cdot 11^{2}
Sign: 0.266−0.963i0.266 - 0.963i
Analytic conductor: 3.864753.86475
Root analytic conductor: 1.965891.96589
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ484(245,⋅)\chi_{484} (245, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 484, ( :1/2), 0.266−0.963i)(2,\ 484,\ (\ :1/2),\ 0.266 - 0.963i)

Particular Values

L(1)L(1) ≈\approx 1.39404+1.06042i1.39404 + 1.06042i
L(12)L(\frac12) ≈\approx 1.39404+1.06042i1.39404 + 1.06042i
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
11 1 1
good3 1+(−0.309−0.951i)T+(−2.42+1.76i)T2 1 + (-0.309 - 0.951i)T + (-2.42 + 1.76i)T^{2}
5 1+(−2.42−1.76i)T+(1.54+4.75i)T2 1 + (-2.42 - 1.76i)T + (1.54 + 4.75i)T^{2}
7 1+(0.618−1.90i)T+(−5.66−4.11i)T2 1 + (0.618 - 1.90i)T + (-5.66 - 4.11i)T^{2}
13 1+(3.23−2.35i)T+(4.01−12.3i)T2 1 + (3.23 - 2.35i)T + (4.01 - 12.3i)T^{2}
17 1+(−4.85−3.52i)T+(5.25+16.1i)T2 1 + (-4.85 - 3.52i)T + (5.25 + 16.1i)T^{2}
19 1+(2.47+7.60i)T+(−15.3+11.1i)T2 1 + (2.47 + 7.60i)T + (-15.3 + 11.1i)T^{2}
23 1+3T+23T2 1 + 3T + 23T^{2}
29 1+(−23.4−17.0i)T2 1 + (-23.4 - 17.0i)T^{2}
31 1+(4.04−2.93i)T+(9.57−29.4i)T2 1 + (4.04 - 2.93i)T + (9.57 - 29.4i)T^{2}
37 1+(0.309−0.951i)T+(−29.9−21.7i)T2 1 + (0.309 - 0.951i)T + (-29.9 - 21.7i)T^{2}
41 1+(−33.1+24.0i)T2 1 + (-33.1 + 24.0i)T^{2}
43 1−10T+43T2 1 - 10T + 43T^{2}
47 1+(−38.0+27.6i)T2 1 + (-38.0 + 27.6i)T^{2}
53 1+(−4.85+3.52i)T+(16.3−50.4i)T2 1 + (-4.85 + 3.52i)T + (16.3 - 50.4i)T^{2}
59 1+(−0.927+2.85i)T+(−47.7−34.6i)T2 1 + (-0.927 + 2.85i)T + (-47.7 - 34.6i)T^{2}
61 1+(3.23+2.35i)T+(18.8+58.0i)T2 1 + (3.23 + 2.35i)T + (18.8 + 58.0i)T^{2}
67 1+T+67T2 1 + T + 67T^{2}
71 1+(12.1+8.81i)T+(21.9+67.5i)T2 1 + (12.1 + 8.81i)T + (21.9 + 67.5i)T^{2}
73 1+(−1.23+3.80i)T+(−59.0−42.9i)T2 1 + (-1.23 + 3.80i)T + (-59.0 - 42.9i)T^{2}
79 1+(−1.61+1.17i)T+(24.4−75.1i)T2 1 + (-1.61 + 1.17i)T + (24.4 - 75.1i)T^{2}
83 1+(−4.85−3.52i)T+(25.6+78.9i)T2 1 + (-4.85 - 3.52i)T + (25.6 + 78.9i)T^{2}
89 1+9T+89T2 1 + 9T + 89T^{2}
97 1+(−5.66+4.11i)T+(29.9−92.2i)T2 1 + (-5.66 + 4.11i)T + (29.9 - 92.2i)T^{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

−10.88494202040169504040875008320, −10.14999554086894116824480479761, −9.481222202858886288920714416068, −8.854729544922470334139443461289, −7.33581250524104030925515002168, −6.49094742748738134341241732196, −5.61727120668016419310224964388, −4.42756399396769183806085103039, −3.08118490892394758431057700956, −2.04171192493874822520990477908, 1.15945484992228854286561639339, 2.35128542778674595783181258781, 4.00517394530586940089465546418, 5.25856857005740610670451872142, 6.01105138310948492920737718366, 7.38957450172641131166273749011, 7.82893649174927800200913964437, 9.113544242314591045984786178084, 10.16289698408393385176577299557, 10.25346058594998907727231018249

Graph of the ZZ-function along the critical line