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Properties

Label 2-35e2-1.1-c1-0-0
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 0.28005940060.2800594006
L(12)L(\frac12) ≈\approx 0.28005940060.2800594006
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

−9.765440959456143660606291540516, −9.168744921377228823570277248462, −8.032271773782449855297453316465, −7.21046475119151909998206313802, −6.48216499272556153360191672892, −5.31098673513379710543099264967, −4.91447632000599858110226606819, −3.88459950169202020847012376724, −1.92657474480101744243828534718, −0.46928745299023079776231010794, 0.46928745299023079776231010794, 1.92657474480101744243828534718, 3.88459950169202020847012376724, 4.91447632000599858110226606819, 5.31098673513379710543099264967, 6.48216499272556153360191672892, 7.21046475119151909998206313802, 8.032271773782449855297453316465, 9.168744921377228823570277248462, 9.765440959456143660606291540516

Graph of the ZZ-function along the critical line