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Properties

Label 2-22e2-1.1-c1-0-1
Degree 22
Conductor 484484
Sign 11
Analytic cond. 3.864753.86475
Root an. cond. 1.965891.96589
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.37·3-s + 4.37·5-s + 2.62·9-s − 10.3·15-s + 1.62·23-s + 14.1·25-s + 0.883·27-s + 11.1·31-s + 5.11·37-s + 11.4·45-s − 12·47-s − 7·49-s + 6·53-s + 10.3·59-s − 15.1·67-s − 3.86·69-s + 15.8·71-s − 33.4·75-s − 9.97·81-s − 9.86·89-s − 26.3·93-s − 17.1·97-s − 4·103-s − 12.1·111-s − 7.62·113-s + 7.11·115-s + ⋯
L(s)  = 1  − 1.36·3-s + 1.95·5-s + 0.875·9-s − 2.67·15-s + 0.339·23-s + 2.82·25-s + 0.169·27-s + 1.99·31-s + 0.841·37-s + 1.71·45-s − 1.75·47-s − 49-s + 0.824·53-s + 1.35·59-s − 1.84·67-s − 0.464·69-s + 1.88·71-s − 3.86·75-s − 1.10·81-s − 1.04·89-s − 2.73·93-s − 1.73·97-s − 0.394·103-s − 1.15·111-s − 0.717·113-s + 0.663·115-s + ⋯

Functional equation

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

Invariants

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

Particular Values

L(1)L(1) ≈\approx 1.2779508901.277950890
L(12)L(\frac12) ≈\approx 1.2779508901.277950890
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+2.37T+3T2 1 + 2.37T + 3T^{2}
5 1−4.37T+5T2 1 - 4.37T + 5T^{2}
7 1+7T2 1 + 7T^{2}
13 1+13T2 1 + 13T^{2}
17 1+17T2 1 + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1−1.62T+23T2 1 - 1.62T + 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1−11.1T+31T2 1 - 11.1T + 31T^{2}
37 1−5.11T+37T2 1 - 5.11T + 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+12T+47T2 1 + 12T + 47T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1−10.3T+59T2 1 - 10.3T + 59T^{2}
61 1+61T2 1 + 61T^{2}
67 1+15.1T+67T2 1 + 15.1T + 67T^{2}
71 1−15.8T+71T2 1 - 15.8T + 71T^{2}
73 1+73T2 1 + 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1+9.86T+89T2 1 + 9.86T + 89T^{2}
97 1+17.1T+97T2 1 + 17.1T + 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

−10.90678716645046167437176099668, −10.09890544132751106411658374752, −9.592596935776882217962835034484, −8.404636360958729750555798476685, −6.75894472225758429941547830740, −6.26167928280954426402695091260, −5.44854738660372947623127224204, −4.72417600974870336732018508119, −2.66755610569428205499615310321, −1.23008901880303198978723571138, 1.23008901880303198978723571138, 2.66755610569428205499615310321, 4.72417600974870336732018508119, 5.44854738660372947623127224204, 6.26167928280954426402695091260, 6.75894472225758429941547830740, 8.404636360958729750555798476685, 9.592596935776882217962835034484, 10.09890544132751106411658374752, 10.90678716645046167437176099668

Graph of the ZZ-function along the critical line