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Properties

Label 2-1078-1.1-c1-0-31
Degree 22
Conductor 10781078
Sign −1-1
Analytic cond. 8.607878.60787
Root an. cond. 2.933912.93391
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2.82·5-s − 8-s − 3·9-s − 2.82·10-s + 11-s − 5.65·13-s + 16-s − 2.82·17-s + 3·18-s − 8.48·19-s + 2.82·20-s − 22-s − 8·23-s + 3.00·25-s + 5.65·26-s − 6·29-s + 8.48·31-s − 32-s + 2.82·34-s − 3·36-s − 6·37-s + 8.48·38-s − 2.82·40-s + 8.48·41-s − 4·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.26·5-s − 0.353·8-s − 9-s − 0.894·10-s + 0.301·11-s − 1.56·13-s + 0.250·16-s − 0.685·17-s + 0.707·18-s − 1.94·19-s + 0.632·20-s − 0.213·22-s − 1.66·23-s + 0.600·25-s + 1.10·26-s − 1.11·29-s + 1.52·31-s − 0.176·32-s + 0.485·34-s − 0.5·36-s − 0.986·37-s + 1.37·38-s − 0.447·40-s + 1.32·41-s − 0.609·43-s + ⋯

Functional equation

Λ(s)=(1078s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1078s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10781078    =    2⋅72⋅112 \cdot 7^{2} \cdot 11
Sign: −1-1
Analytic conductor: 8.607878.60787
Root analytic conductor: 2.933912.93391
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1078, ( :1/2), −1)(2,\ 1078,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+T 1 + T
7 1 1
11 1−T 1 - T
good3 1+3T2 1 + 3T^{2}
5 1−2.82T+5T2 1 - 2.82T + 5T^{2}
13 1+5.65T+13T2 1 + 5.65T + 13T^{2}
17 1+2.82T+17T2 1 + 2.82T + 17T^{2}
19 1+8.48T+19T2 1 + 8.48T + 19T^{2}
23 1+8T+23T2 1 + 8T + 23T^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1−8.48T+31T2 1 - 8.48T + 31T^{2}
37 1+6T+37T2 1 + 6T + 37T^{2}
41 1−8.48T+41T2 1 - 8.48T + 41T^{2}
43 1+4T+43T2 1 + 4T + 43T^{2}
47 1−2.82T+47T2 1 - 2.82T + 47T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1−5.65T+59T2 1 - 5.65T + 59T^{2}
61 1−5.65T+61T2 1 - 5.65T + 61T^{2}
67 1+4T+67T2 1 + 4T + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−8.48T+73T2 1 - 8.48T + 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−2.82T+83T2 1 - 2.82T + 83T^{2}
89 1+11.3T+89T2 1 + 11.3T + 89T^{2}
97 1+11.3T+97T2 1 + 11.3T + 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

−9.530730185409156455610738588161, −8.734877997248979398883447111660, −8.039633612488981166741343372411, −6.85101474999731051432473448282, −6.17053049575869876704436871049, −5.42267652915547103492609807415, −4.18712210120360699910405087214, −2.46348138242626626833794123960, −2.08526057766359163412999320943, 0, 2.08526057766359163412999320943, 2.46348138242626626833794123960, 4.18712210120360699910405087214, 5.42267652915547103492609807415, 6.17053049575869876704436871049, 6.85101474999731051432473448282, 8.039633612488981166741343372411, 8.734877997248979398883447111660, 9.530730185409156455610738588161

Graph of the ZZ-function along the critical line