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Properties

Label 2-2548-1.1-c1-0-1
Degree 22
Conductor 25482548
Sign 11
Analytic cond. 20.345820.3458
Root an. cond. 4.510644.51064
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  − 2.44·3-s − 1.44·5-s + 2.99·9-s + 1.55·11-s + 13-s + 3.55·15-s + 2.44·17-s − 5.44·19-s − 5.89·23-s − 2.89·25-s + 3.89·29-s + 1.44·31-s − 3.79·33-s − 3.55·37-s − 2.44·39-s − 1.10·41-s + 43-s − 4.34·45-s + 1.44·47-s − 5.99·51-s − 7.89·53-s − 2.24·55-s + 13.3·57-s − 14·59-s + 2·61-s − 1.44·65-s + 2.89·67-s + ⋯
L(s)  = 1  − 1.41·3-s − 0.648·5-s + 0.999·9-s + 0.467·11-s + 0.277·13-s + 0.916·15-s + 0.594·17-s − 1.25·19-s − 1.23·23-s − 0.579·25-s + 0.724·29-s + 0.260·31-s − 0.661·33-s − 0.583·37-s − 0.392·39-s − 0.171·41-s + 0.152·43-s − 0.648·45-s + 0.211·47-s − 0.840·51-s − 1.08·53-s − 0.303·55-s + 1.76·57-s − 1.82·59-s + 0.256·61-s − 0.179·65-s + 0.354·67-s + ⋯

Functional equation

Λ(s)=(2548s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2548s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25482548    =    22⋅72⋅132^{2} \cdot 7^{2} \cdot 13
Sign: 11
Analytic conductor: 20.345820.3458
Root analytic conductor: 4.510644.51064
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2548, ( :1/2), 1)(2,\ 2548,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.66597446220.6659744622
L(12)L(\frac12) ≈\approx 0.66597446220.6659744622
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
13 1−T 1 - T
good3 1+2.44T+3T2 1 + 2.44T + 3T^{2}
5 1+1.44T+5T2 1 + 1.44T + 5T^{2}
11 1−1.55T+11T2 1 - 1.55T + 11T^{2}
17 1−2.44T+17T2 1 - 2.44T + 17T^{2}
19 1+5.44T+19T2 1 + 5.44T + 19T^{2}
23 1+5.89T+23T2 1 + 5.89T + 23T^{2}
29 1−3.89T+29T2 1 - 3.89T + 29T^{2}
31 1−1.44T+31T2 1 - 1.44T + 31T^{2}
37 1+3.55T+37T2 1 + 3.55T + 37T^{2}
41 1+1.10T+41T2 1 + 1.10T + 41T^{2}
43 1−T+43T2 1 - T + 43T^{2}
47 1−1.44T+47T2 1 - 1.44T + 47T^{2}
53 1+7.89T+53T2 1 + 7.89T + 53T^{2}
59 1+14T+59T2 1 + 14T + 59T^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1−2.89T+67T2 1 - 2.89T + 67T^{2}
71 1−7.55T+71T2 1 - 7.55T + 71T^{2}
73 1−13.2T+73T2 1 - 13.2T + 73T^{2}
79 1+11.8T+79T2 1 + 11.8T + 79T^{2}
83 1+10.3T+83T2 1 + 10.3T + 83T^{2}
89 1−6.55T+89T2 1 - 6.55T + 89T^{2}
97 1−15.4T+97T2 1 - 15.4T + 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

−8.798772106184225285536196335381, −8.081538455527208371188214947018, −7.27922739571828070743884488593, −6.24018790693486286306002051263, −6.09266770036734299740367835407, −4.93761000827692571057122557010, −4.28186671022114994007035871071, −3.41373726259073871327280850401, −1.87333330443159406517556187339, −0.54986255372974326845782942662, 0.54986255372974326845782942662, 1.87333330443159406517556187339, 3.41373726259073871327280850401, 4.28186671022114994007035871071, 4.93761000827692571057122557010, 6.09266770036734299740367835407, 6.24018790693486286306002051263, 7.27922739571828070743884488593, 8.081538455527208371188214947018, 8.798772106184225285536196335381

Graph of the ZZ-function along the critical line