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Properties

Label 2-2535-1.1-c1-0-96
Degree 22
Conductor 25352535
Sign −1-1
Analytic cond. 20.242020.2420
Root an. cond. 4.499114.49911
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  + 1.24·2-s + 3-s − 0.445·4-s − 5-s + 1.24·6-s − 0.246·7-s − 3.04·8-s + 9-s − 1.24·10-s + 0.109·11-s − 0.445·12-s − 0.307·14-s − 15-s − 2.91·16-s − 1.50·17-s + 1.24·18-s + 3.44·19-s + 0.445·20-s − 0.246·21-s + 0.137·22-s − 7.80·23-s − 3.04·24-s + 25-s + 27-s + 0.109·28-s − 4.66·29-s − 1.24·30-s + ⋯
L(s)  = 1  + 0.881·2-s + 0.577·3-s − 0.222·4-s − 0.447·5-s + 0.509·6-s − 0.0933·7-s − 1.07·8-s + 0.333·9-s − 0.394·10-s + 0.0331·11-s − 0.128·12-s − 0.0823·14-s − 0.258·15-s − 0.727·16-s − 0.365·17-s + 0.293·18-s + 0.790·19-s + 0.0995·20-s − 0.0538·21-s + 0.0292·22-s − 1.62·23-s − 0.622·24-s + 0.200·25-s + 0.192·27-s + 0.0207·28-s − 0.866·29-s − 0.227·30-s + ⋯

Functional equation

Λ(s)=(2535s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2535s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25352535    =    3⋅5⋅1323 \cdot 5 \cdot 13^{2}
Sign: −1-1
Analytic conductor: 20.242020.2420
Root analytic conductor: 4.499114.49911
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2535, ( :1/2), −1)(2,\ 2535,\ (\ :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)
bad3 1−T 1 - T
5 1+T 1 + T
13 1 1
good2 1−1.24T+2T2 1 - 1.24T + 2T^{2}
7 1+0.246T+7T2 1 + 0.246T + 7T^{2}
11 1−0.109T+11T2 1 - 0.109T + 11T^{2}
17 1+1.50T+17T2 1 + 1.50T + 17T^{2}
19 1−3.44T+19T2 1 - 3.44T + 19T^{2}
23 1+7.80T+23T2 1 + 7.80T + 23T^{2}
29 1+4.66T+29T2 1 + 4.66T + 29T^{2}
31 1−3.27T+31T2 1 - 3.27T + 31T^{2}
37 1−0.939T+37T2 1 - 0.939T + 37T^{2}
41 1+4.24T+41T2 1 + 4.24T + 41T^{2}
43 1+11.2T+43T2 1 + 11.2T + 43T^{2}
47 1+11.4T+47T2 1 + 11.4T + 47T^{2}
53 1+7.34T+53T2 1 + 7.34T + 53T^{2}
59 1+3.46T+59T2 1 + 3.46T + 59T^{2}
61 1+0.978T+61T2 1 + 0.978T + 61T^{2}
67 1+10.5T+67T2 1 + 10.5T + 67T^{2}
71 1−7.09T+71T2 1 - 7.09T + 71T^{2}
73 1−3.44T+73T2 1 - 3.44T + 73T^{2}
79 1−9.27T+79T2 1 - 9.27T + 79T^{2}
83 1−7.92T+83T2 1 - 7.92T + 83T^{2}
89 1−1.62T+89T2 1 - 1.62T + 89T^{2}
97 1+5.81T+97T2 1 + 5.81T + 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

−8.260041205750820050298404835106, −8.031503479119975708216320310254, −6.83807996402292084023526942428, −6.15365676067489718953062717948, −5.14967863719663077994464267563, −4.48739965478170219925189236143, −3.61164013062978112014972415273, −3.09431259761080525220552955485, −1.81083073995611742153847986908, 0, 1.81083073995611742153847986908, 3.09431259761080525220552955485, 3.61164013062978112014972415273, 4.48739965478170219925189236143, 5.14967863719663077994464267563, 6.15365676067489718953062717948, 6.83807996402292084023526942428, 8.031503479119975708216320310254, 8.260041205750820050298404835106

Graph of the ZZ-function along the critical line