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Properties

Label 2-22e2-11.4-c1-0-7
Degree 22
Conductor 484484
Sign −0.975+0.220i-0.975 + 0.220i
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  + (0.618 − 1.90i)3-s + (−2.42 + 1.76i)5-s + (−1.07 − 3.29i)7-s + (−0.809 − 0.587i)9-s + (−4.20 − 3.05i)13-s + (1.85 + 5.70i)15-s + (−4.20 + 3.05i)17-s + (1.07 − 3.29i)19-s − 6.92·21-s − 6·23-s + (1.23 − 3.80i)25-s + (3.23 − 2.35i)27-s + (−1.60 − 4.94i)29-s + (1.61 + 1.17i)31-s + (8.40 + 6.10i)35-s + ⋯
L(s)  = 1  + (0.356 − 1.09i)3-s + (−1.08 + 0.788i)5-s + (−0.404 − 1.24i)7-s + (−0.269 − 0.195i)9-s + (−1.16 − 0.847i)13-s + (0.478 + 1.47i)15-s + (−1.01 + 0.740i)17-s + (0.245 − 0.755i)19-s − 1.51·21-s − 1.25·23-s + (0.247 − 0.760i)25-s + (0.622 − 0.452i)27-s + (−0.298 − 0.917i)29-s + (0.290 + 0.211i)31-s + (1.42 + 1.03i)35-s + ⋯

Functional equation

Λ(s)=(484s/2 ΓC(s) L(s)=((−0.975+0.220i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(484s/2 ΓC(s+1/2) L(s)=((−0.975+0.220i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 484484    =    22⋅1122^{2} \cdot 11^{2}
Sign: −0.975+0.220i-0.975 + 0.220i
Analytic conductor: 3.864753.86475
Root analytic conductor: 1.965891.96589
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ484(81,⋅)\chi_{484} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 484, ( :1/2), −0.975+0.220i)(2,\ 484,\ (\ :1/2),\ -0.975 + 0.220i)

Particular Values

L(1)L(1) ≈\approx 0.0728288−0.651908i0.0728288 - 0.651908i
L(12)L(\frac12) ≈\approx 0.0728288−0.651908i0.0728288 - 0.651908i
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+(−0.618+1.90i)T+(−2.42−1.76i)T2 1 + (-0.618 + 1.90i)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+(1.07+3.29i)T+(−5.66+4.11i)T2 1 + (1.07 + 3.29i)T + (-5.66 + 4.11i)T^{2}
13 1+(4.20+3.05i)T+(4.01+12.3i)T2 1 + (4.20 + 3.05i)T + (4.01 + 12.3i)T^{2}
17 1+(4.20−3.05i)T+(5.25−16.1i)T2 1 + (4.20 - 3.05i)T + (5.25 - 16.1i)T^{2}
19 1+(−1.07+3.29i)T+(−15.3−11.1i)T2 1 + (-1.07 + 3.29i)T + (-15.3 - 11.1i)T^{2}
23 1+6T+23T2 1 + 6T + 23T^{2}
29 1+(1.60+4.94i)T+(−23.4+17.0i)T2 1 + (1.60 + 4.94i)T + (-23.4 + 17.0i)T^{2}
31 1+(−1.61−1.17i)T+(9.57+29.4i)T2 1 + (-1.61 - 1.17i)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+(1.60−4.94i)T+(−33.1−24.0i)T2 1 + (1.60 - 4.94i)T + (-33.1 - 24.0i)T^{2}
43 1+43T2 1 + 43T^{2}
47 1+(−1.85+5.70i)T+(−38.0−27.6i)T2 1 + (-1.85 + 5.70i)T + (-38.0 - 27.6i)T^{2}
53 1+(−2.42−1.76i)T+(16.3+50.4i)T2 1 + (-2.42 - 1.76i)T + (16.3 + 50.4i)T^{2}
59 1+(−47.7+34.6i)T2 1 + (-47.7 + 34.6i)T^{2}
61 1+(−11.2+8.14i)T+(18.8−58.0i)T2 1 + (-11.2 + 8.14i)T + (18.8 - 58.0i)T^{2}
67 1+2T+67T2 1 + 2T + 67T^{2}
71 1+(21.9−67.5i)T2 1 + (21.9 - 67.5i)T^{2}
73 1+(2.14+6.58i)T+(−59.0+42.9i)T2 1 + (2.14 + 6.58i)T + (-59.0 + 42.9i)T^{2}
79 1+(2.80+2.03i)T+(24.4+75.1i)T2 1 + (2.80 + 2.03i)T + (24.4 + 75.1i)T^{2}
83 1+(−8.40+6.10i)T+(25.6−78.9i)T2 1 + (-8.40 + 6.10i)T + (25.6 - 78.9i)T^{2}
89 1−15T+89T2 1 - 15T + 89T^{2}
97 1+(4.04+2.93i)T+(29.9+92.2i)T2 1 + (4.04 + 2.93i)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

−10.57452885704478619954083217910, −9.902811524831733221658861561031, −8.321831380003533144693634970550, −7.61927707829042791593551834654, −7.13977268665294810624635897971, −6.36523027303624625484282957568, −4.53866992888033644827080646767, −3.52235981530607976794156224896, −2.32333578395404876215973629464, −0.36148131048104681031669543136, 2.43121871883488838962939537375, 3.80524034706204569090963199660, 4.55101552156429345134867232519, 5.44528180839110309185690222641, 6.88978409245489273468400204120, 8.050559493354519739151629485784, 8.999547538451491234567072430202, 9.344036891858636104214463272551, 10.30804182061234352392479610720, 11.64563345249898489616272164366

Graph of the ZZ-function along the critical line