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Properties

Label 2-1127-1.1-c1-0-29
Degree 22
Conductor 11271127
Sign −1-1
Analytic cond. 8.999148.99914
Root an. cond. 2.999852.99985
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  − 0.134·2-s − 2.58·3-s − 1.98·4-s − 1.71·5-s + 0.348·6-s + 0.536·8-s + 3.68·9-s + 0.231·10-s + 3.43·11-s + 5.12·12-s − 0.592·13-s + 4.43·15-s + 3.89·16-s + 0.811·17-s − 0.496·18-s − 1.38·19-s + 3.39·20-s − 0.463·22-s − 23-s − 1.38·24-s − 2.06·25-s + 0.0798·26-s − 1.76·27-s + 6.39·29-s − 0.597·30-s + 7.03·31-s − 1.59·32-s + ⋯
L(s)  = 1  − 0.0953·2-s − 1.49·3-s − 0.990·4-s − 0.766·5-s + 0.142·6-s + 0.189·8-s + 1.22·9-s + 0.0731·10-s + 1.03·11-s + 1.47·12-s − 0.164·13-s + 1.14·15-s + 0.972·16-s + 0.196·17-s − 0.117·18-s − 0.318·19-s + 0.759·20-s − 0.0987·22-s − 0.208·23-s − 0.283·24-s − 0.412·25-s + 0.0156·26-s − 0.340·27-s + 1.18·29-s − 0.109·30-s + 1.26·31-s − 0.282·32-s + ⋯

Functional equation

Λ(s)=(1127s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1127s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11271127    =    72⋅237^{2} \cdot 23
Sign: −1-1
Analytic conductor: 8.999148.99914
Root analytic conductor: 2.999852.99985
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1127, ( :1/2), −1)(2,\ 1127,\ (\ :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)
bad7 1 1
23 1+T 1 + T
good2 1+0.134T+2T2 1 + 0.134T + 2T^{2}
3 1+2.58T+3T2 1 + 2.58T + 3T^{2}
5 1+1.71T+5T2 1 + 1.71T + 5T^{2}
11 1−3.43T+11T2 1 - 3.43T + 11T^{2}
13 1+0.592T+13T2 1 + 0.592T + 13T^{2}
17 1−0.811T+17T2 1 - 0.811T + 17T^{2}
19 1+1.38T+19T2 1 + 1.38T + 19T^{2}
29 1−6.39T+29T2 1 - 6.39T + 29T^{2}
31 1−7.03T+31T2 1 - 7.03T + 31T^{2}
37 1−10.3T+37T2 1 - 10.3T + 37T^{2}
41 1+7.45T+41T2 1 + 7.45T + 41T^{2}
43 1+8.85T+43T2 1 + 8.85T + 43T^{2}
47 1+10.5T+47T2 1 + 10.5T + 47T^{2}
53 1+11.8T+53T2 1 + 11.8T + 53T^{2}
59 1+0.998T+59T2 1 + 0.998T + 59T^{2}
61 1+7.86T+61T2 1 + 7.86T + 61T^{2}
67 1−3.79T+67T2 1 - 3.79T + 67T^{2}
71 1−10.0T+71T2 1 - 10.0T + 71T^{2}
73 1−0.224T+73T2 1 - 0.224T + 73T^{2}
79 1+7.18T+79T2 1 + 7.18T + 79T^{2}
83 1+4.71T+83T2 1 + 4.71T + 83T^{2}
89 1−8.59T+89T2 1 - 8.59T + 89T^{2}
97 1+8.38T+97T2 1 + 8.38T + 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

−9.667373652132272407503083914844, −8.498501737567128101633243800457, −7.87093153884400311122941323774, −6.62648062020478851624477710905, −6.10929169828156576902101610337, −4.87144124140411595962348644561, −4.49811659001842224459152397090, −3.42912668460487221769741071536, −1.19048916826252840745734959389, 0, 1.19048916826252840745734959389, 3.42912668460487221769741071536, 4.49811659001842224459152397090, 4.87144124140411595962348644561, 6.10929169828156576902101610337, 6.62648062020478851624477710905, 7.87093153884400311122941323774, 8.498501737567128101633243800457, 9.667373652132272407503083914844

Graph of the ZZ-function along the critical line