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Properties

Label 2-35e2-1.1-c1-0-28
Degree 22
Conductor 12251225
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.23·3-s − 2·4-s + 2.00·9-s − 3·11-s + 4.47·12-s + 6.70·13-s + 4·16-s + 2.23·17-s + 2.23·27-s − 9·29-s + 6.70·33-s − 4.00·36-s − 15.0·39-s + 6·44-s − 11.1·47-s − 8.94·48-s − 5.00·51-s − 13.4·52-s − 8·64-s − 4.47·68-s − 12·71-s + 13.4·73-s − 79-s − 11·81-s − 8.94·83-s + 20.1·87-s − 6.70·97-s + ⋯
L(s)  = 1  − 1.29·3-s − 4-s + 0.666·9-s − 0.904·11-s + 1.29·12-s + 1.86·13-s + 16-s + 0.542·17-s + 0.430·27-s − 1.67·29-s + 1.16·33-s − 0.666·36-s − 2.40·39-s + 0.904·44-s − 1.63·47-s − 1.29·48-s − 0.700·51-s − 1.86·52-s − 64-s − 0.542·68-s − 1.42·71-s + 1.57·73-s − 0.112·79-s − 1.22·81-s − 0.981·83-s + 2.15·87-s − 0.681·97-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: −1-1
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: 11
Selberg data: (2, 1225, ( :1/2), −1)(2,\ 1225,\ (\ :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
7 1 1
good2 1+2T2 1 + 2T^{2}
3 1+2.23T+3T2 1 + 2.23T + 3T^{2}
11 1+3T+11T2 1 + 3T + 11T^{2}
13 1−6.70T+13T2 1 - 6.70T + 13T^{2}
17 1−2.23T+17T2 1 - 2.23T + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1+9T+29T2 1 + 9T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+37T2 1 + 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+11.1T+47T2 1 + 11.1T + 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+61T2 1 + 61T^{2}
67 1+67T2 1 + 67T^{2}
71 1+12T+71T2 1 + 12T + 71T^{2}
73 1−13.4T+73T2 1 - 13.4T + 73T^{2}
79 1+T+79T2 1 + T + 79T^{2}
83 1+8.94T+83T2 1 + 8.94T + 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1+6.70T+97T2 1 + 6.70T + 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

−9.388184580540948124959442289420, −8.479529323929461630967714085177, −7.80297977430087953436193082919, −6.56743779994051427602049302435, −5.67600780303091926205299634625, −5.32168678165937101325745264997, −4.23446721686868124359633520296, −3.30678461771693015492849077192, −1.30915866113986890771182369008, 0, 1.30915866113986890771182369008, 3.30678461771693015492849077192, 4.23446721686868124359633520296, 5.32168678165937101325745264997, 5.67600780303091926205299634625, 6.56743779994051427602049302435, 7.80297977430087953436193082919, 8.479529323929461630967714085177, 9.388184580540948124959442289420

Graph of the ZZ-function along the critical line