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Properties

Label 2-1338-1.1-c1-0-19
Degree 22
Conductor 13381338
Sign 11
Analytic cond. 10.683910.6839
Root an. cond. 3.268633.26863
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 + 3-s + 4-s + 0.137·5-s + 6-s + 3.14·7-s + 8-s + 9-s + 0.137·10-s + 1.47·11-s + 12-s − 1.75·13-s + 3.14·14-s + 0.137·15-s + 16-s − 4.07·17-s + 18-s + 1.58·19-s + 0.137·20-s + 3.14·21-s + 1.47·22-s + 8.54·23-s + 24-s − 4.98·25-s − 1.75·26-s + 27-s + 3.14·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.0613·5-s + 0.408·6-s + 1.18·7-s + 0.353·8-s + 0.333·9-s + 0.0433·10-s + 0.445·11-s + 0.288·12-s − 0.487·13-s + 0.839·14-s + 0.0354·15-s + 0.250·16-s − 0.989·17-s + 0.235·18-s + 0.364·19-s + 0.0306·20-s + 0.685·21-s + 0.314·22-s + 1.78·23-s + 0.204·24-s − 0.996·25-s − 0.344·26-s + 0.192·27-s + 0.593·28-s + ⋯

Functional equation

Λ(s)=(1338s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1338s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13381338    =    2⋅3⋅2232 \cdot 3 \cdot 223
Sign: 11
Analytic conductor: 10.683910.6839
Root analytic conductor: 3.268633.26863
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1338, ( :1/2), 1)(2,\ 1338,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.6627945293.662794529
L(12)L(\frac12) ≈\approx 3.6627945293.662794529
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−T 1 - T
223 1−T 1 - T
good5 1−0.137T+5T2 1 - 0.137T + 5T^{2}
7 1−3.14T+7T2 1 - 3.14T + 7T^{2}
11 1−1.47T+11T2 1 - 1.47T + 11T^{2}
13 1+1.75T+13T2 1 + 1.75T + 13T^{2}
17 1+4.07T+17T2 1 + 4.07T + 17T^{2}
19 1−1.58T+19T2 1 - 1.58T + 19T^{2}
23 1−8.54T+23T2 1 - 8.54T + 23T^{2}
29 1−1.49T+29T2 1 - 1.49T + 29T^{2}
31 1−0.711T+31T2 1 - 0.711T + 31T^{2}
37 1+6.13T+37T2 1 + 6.13T + 37T^{2}
41 1−4.46T+41T2 1 - 4.46T + 41T^{2}
43 1+4.37T+43T2 1 + 4.37T + 43T^{2}
47 1−3.50T+47T2 1 - 3.50T + 47T^{2}
53 1+8.57T+53T2 1 + 8.57T + 53T^{2}
59 1+0.0537T+59T2 1 + 0.0537T + 59T^{2}
61 1+13.6T+61T2 1 + 13.6T + 61T^{2}
67 1−6.84T+67T2 1 - 6.84T + 67T^{2}
71 1+1.11T+71T2 1 + 1.11T + 71T^{2}
73 1−7.44T+73T2 1 - 7.44T + 73T^{2}
79 1+5.85T+79T2 1 + 5.85T + 79T^{2}
83 1−8.76T+83T2 1 - 8.76T + 83T^{2}
89 1−13.7T+89T2 1 - 13.7T + 89T^{2}
97 1+2.93T+97T2 1 + 2.93T + 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

−9.482151478271728627546117509300, −8.797072599136514013813654424228, −7.894329022570576977701153824573, −7.20404994181485955035389662166, −6.33758118977288126870524415542, −5.10118056948758892882713869020, −4.62173370353024827775953940146, −3.56354993593256664956988032635, −2.46649901818912267667727627398, −1.47676215755220921337298441505, 1.47676215755220921337298441505, 2.46649901818912267667727627398, 3.56354993593256664956988032635, 4.62173370353024827775953940146, 5.10118056948758892882713869020, 6.33758118977288126870524415542, 7.20404994181485955035389662166, 7.894329022570576977701153824573, 8.797072599136514013813654424228, 9.482151478271728627546117509300

Graph of the ZZ-function along the critical line