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Properties

Label 2-35e2-1.1-c1-0-2
Degree 22
Conductor 12251225
Sign 11
Analytic cond. 9.781679.78167
Root an. cond. 3.127563.12756
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-s − 2.82·3-s − 4-s − 2.82·6-s − 3·8-s + 5.00·9-s + 2.82·12-s − 4.24·13-s − 16-s − 4.24·17-s + 5.00·18-s − 2.82·19-s + 4·23-s + 8.48·24-s − 4.24·26-s − 5.65·27-s + 5.65·31-s + 5·32-s − 4.24·34-s − 5.00·36-s + 6·37-s − 2.82·38-s + 12·39-s + 4.24·41-s + 4·46-s + 2.82·48-s + 12·51-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.63·3-s − 0.5·4-s − 1.15·6-s − 1.06·8-s + 1.66·9-s + 0.816·12-s − 1.17·13-s − 0.250·16-s − 1.02·17-s + 1.17·18-s − 0.648·19-s + 0.834·23-s + 1.73·24-s − 0.832·26-s − 1.08·27-s + 1.01·31-s + 0.883·32-s − 0.727·34-s − 0.833·36-s + 0.986·37-s − 0.458·38-s + 1.92·39-s + 0.662·41-s + 0.589·46-s + 0.408·48-s + 1.68·51-s + ⋯

Functional equation

Λ(s)=(1225s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1225s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12251225    =    52⋅725^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 9.781679.78167
Root analytic conductor: 3.127563.12756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1225, ( :1/2), 1)(2,\ 1225,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.73088326520.7308832652
L(12)L(\frac12) ≈\approx 0.73088326520.7308832652
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
7 1 1
good2 1−T+2T2 1 - T + 2T^{2}
3 1+2.82T+3T2 1 + 2.82T + 3T^{2}
11 1+11T2 1 + 11T^{2}
13 1+4.24T+13T2 1 + 4.24T + 13T^{2}
17 1+4.24T+17T2 1 + 4.24T + 17T^{2}
19 1+2.82T+19T2 1 + 2.82T + 19T^{2}
23 1−4T+23T2 1 - 4T + 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1−5.65T+31T2 1 - 5.65T + 31T^{2}
37 1−6T+37T2 1 - 6T + 37T^{2}
41 1−4.24T+41T2 1 - 4.24T + 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1+8T+53T2 1 + 8T + 53T^{2}
59 1−8.48T+59T2 1 - 8.48T + 59T^{2}
61 1−9.89T+61T2 1 - 9.89T + 61T^{2}
67 1+12T+67T2 1 + 12T + 67T^{2}
71 1−12T+71T2 1 - 12T + 71T^{2}
73 1−12.7T+73T2 1 - 12.7T + 73T^{2}
79 1−12T+79T2 1 - 12T + 79T^{2}
83 1−8.48T+83T2 1 - 8.48T + 83T^{2}
89 1+4.24T+89T2 1 + 4.24T + 89T^{2}
97 1−4.24T+97T2 1 - 4.24T + 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

−9.837994967643485875386341524897, −9.120243484267251409314341815232, −7.969689087758694515812449954851, −6.75816224192841022856651307800, −6.30078858233938875674405760206, −5.26845740515687726986433825681, −4.79446893628447739625188208906, −4.06548061239794633931168717861, −2.53909939841841286474128470107, −0.60370243124921022818705555720, 0.60370243124921022818705555720, 2.53909939841841286474128470107, 4.06548061239794633931168717861, 4.79446893628447739625188208906, 5.26845740515687726986433825681, 6.30078858233938875674405760206, 6.75816224192841022856651307800, 7.969689087758694515812449954851, 9.120243484267251409314341815232, 9.837994967643485875386341524897

Graph of the ZZ-function along the critical line