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Properties

Label 2-2646-1.1-c1-0-48
Degree 22
Conductor 26462646
Sign −1-1
Analytic cond. 21.128421.1284
Root an. cond. 4.596564.59656
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 − 1.41·5-s + 8-s − 1.41·10-s + 2.24·11-s − 3·13-s + 16-s − 4.41·17-s − 4.58·19-s − 1.41·20-s + 2.24·22-s − 23-s − 2.99·25-s − 3·26-s − 5.24·29-s + 1.24·31-s + 32-s − 4.41·34-s − 10.4·37-s − 4.58·38-s − 1.41·40-s + 2.82·41-s + 3.24·43-s + 2.24·44-s − 46-s + 7.07·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.632·5-s + 0.353·8-s − 0.447·10-s + 0.676·11-s − 0.832·13-s + 0.250·16-s − 1.07·17-s − 1.05·19-s − 0.316·20-s + 0.478·22-s − 0.208·23-s − 0.599·25-s − 0.588·26-s − 0.973·29-s + 0.223·31-s + 0.176·32-s − 0.757·34-s − 1.72·37-s − 0.743·38-s − 0.223·40-s + 0.441·41-s + 0.494·43-s + 0.338·44-s − 0.147·46-s + 1.03·47-s + ⋯

Functional equation

Λ(s)=(2646s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2646s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26462646    =    2⋅33⋅722 \cdot 3^{3} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 21.128421.1284
Root analytic conductor: 4.596564.59656
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2646, ( :1/2), −1)(2,\ 2646,\ (\ :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
3 1 1
7 1 1
good5 1+1.41T+5T2 1 + 1.41T + 5T^{2}
11 1−2.24T+11T2 1 - 2.24T + 11T^{2}
13 1+3T+13T2 1 + 3T + 13T^{2}
17 1+4.41T+17T2 1 + 4.41T + 17T^{2}
19 1+4.58T+19T2 1 + 4.58T + 19T^{2}
23 1+T+23T2 1 + T + 23T^{2}
29 1+5.24T+29T2 1 + 5.24T + 29T^{2}
31 1−1.24T+31T2 1 - 1.24T + 31T^{2}
37 1+10.4T+37T2 1 + 10.4T + 37T^{2}
41 1−2.82T+41T2 1 - 2.82T + 41T^{2}
43 1−3.24T+43T2 1 - 3.24T + 43T^{2}
47 1−7.07T+47T2 1 - 7.07T + 47T^{2}
53 1−11.2T+53T2 1 - 11.2T + 53T^{2}
59 1+0.171T+59T2 1 + 0.171T + 59T^{2}
61 1+11.6T+61T2 1 + 11.6T + 61T^{2}
67 1−13.2T+67T2 1 - 13.2T + 67T^{2}
71 1+5.48T+71T2 1 + 5.48T + 71T^{2}
73 1+9.89T+73T2 1 + 9.89T + 73T^{2}
79 1+10.7T+79T2 1 + 10.7T + 79T^{2}
83 1+17.3T+83T2 1 + 17.3T + 83T^{2}
89 1−1.92T+89T2 1 - 1.92T + 89T^{2}
97 1+14.8T+97T2 1 + 14.8T + 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

−8.500793991931724638653489871972, −7.46735537077015873032142628226, −6.99050028109064832252254631565, −6.13500277211491840435553766669, −5.30192695990541561140331364463, −4.21616604767026033961272550955, −3.99537247150537430840121571645, −2.71560876310166327303284802940, −1.80220743335325392270577860498, 0, 1.80220743335325392270577860498, 2.71560876310166327303284802940, 3.99537247150537430840121571645, 4.21616604767026033961272550955, 5.30192695990541561140331364463, 6.13500277211491840435553766669, 6.99050028109064832252254631565, 7.46735537077015873032142628226, 8.500793991931724638653489871972

Graph of the ZZ-function along the critical line