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Properties

Label 2-4925-1.1-c1-0-242
Degree 22
Conductor 49254925
Sign −1-1
Analytic cond. 39.326339.3263
Root an. cond. 6.271076.27107
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  − 0.949·2-s + 1.30·3-s − 1.09·4-s − 1.23·6-s + 1.82·7-s + 2.94·8-s − 1.30·9-s + 3.24·11-s − 1.42·12-s + 1.15·13-s − 1.73·14-s − 0.596·16-s − 3.53·17-s + 1.24·18-s − 2.07·19-s + 2.37·21-s − 3.07·22-s + 2.93·23-s + 3.82·24-s − 1.09·26-s − 5.60·27-s − 2.00·28-s − 5.23·29-s − 10.0·31-s − 5.31·32-s + 4.21·33-s + 3.35·34-s + ⋯
L(s)  = 1  − 0.671·2-s + 0.750·3-s − 0.549·4-s − 0.504·6-s + 0.691·7-s + 1.04·8-s − 0.436·9-s + 0.977·11-s − 0.412·12-s + 0.320·13-s − 0.464·14-s − 0.149·16-s − 0.856·17-s + 0.293·18-s − 0.476·19-s + 0.518·21-s − 0.656·22-s + 0.612·23-s + 0.780·24-s − 0.215·26-s − 1.07·27-s − 0.379·28-s − 0.972·29-s − 1.79·31-s − 0.940·32-s + 0.734·33-s + 0.575·34-s + ⋯

Functional equation

Λ(s)=(4925s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4925s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 49254925    =    52⋅1975^{2} \cdot 197
Sign: −1-1
Analytic conductor: 39.326339.3263
Root analytic conductor: 6.271076.27107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4925, ( :1/2), −1)(2,\ 4925,\ (\ :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)
bad5 1 1
197 1−T 1 - T
good2 1+0.949T+2T2 1 + 0.949T + 2T^{2}
3 1−1.30T+3T2 1 - 1.30T + 3T^{2}
7 1−1.82T+7T2 1 - 1.82T + 7T^{2}
11 1−3.24T+11T2 1 - 3.24T + 11T^{2}
13 1−1.15T+13T2 1 - 1.15T + 13T^{2}
17 1+3.53T+17T2 1 + 3.53T + 17T^{2}
19 1+2.07T+19T2 1 + 2.07T + 19T^{2}
23 1−2.93T+23T2 1 - 2.93T + 23T^{2}
29 1+5.23T+29T2 1 + 5.23T + 29T^{2}
31 1+10.0T+31T2 1 + 10.0T + 31T^{2}
37 1−5.29T+37T2 1 - 5.29T + 37T^{2}
41 1+10.0T+41T2 1 + 10.0T + 41T^{2}
43 1−1.09T+43T2 1 - 1.09T + 43T^{2}
47 1−1.47T+47T2 1 - 1.47T + 47T^{2}
53 1+8.57T+53T2 1 + 8.57T + 53T^{2}
59 1−3.47T+59T2 1 - 3.47T + 59T^{2}
61 1+1.91T+61T2 1 + 1.91T + 61T^{2}
67 1−1.85T+67T2 1 - 1.85T + 67T^{2}
71 1−10.5T+71T2 1 - 10.5T + 71T^{2}
73 1+0.875T+73T2 1 + 0.875T + 73T^{2}
79 1+9.47T+79T2 1 + 9.47T + 79T^{2}
83 1+2.08T+83T2 1 + 2.08T + 83T^{2}
89 1−6.90T+89T2 1 - 6.90T + 89T^{2}
97 1+8.32T+97T2 1 + 8.32T + 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

−8.183826695247824301430980530389, −7.43776229751824787481869474325, −6.67570888845430898188827482689, −5.66522309139040204204242296509, −4.85677318767974979936990592958, −4.03636015946896460209135731808, −3.41941659965215272570252974901, −2.12182768425154856188706450323, −1.43556047293116941897070793564, 0, 1.43556047293116941897070793564, 2.12182768425154856188706450323, 3.41941659965215272570252974901, 4.03636015946896460209135731808, 4.85677318767974979936990592958, 5.66522309139040204204242296509, 6.67570888845430898188827482689, 7.43776229751824787481869474325, 8.183826695247824301430980530389

Graph of the ZZ-function along the critical line