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Properties

Label 2-1815-1.1-c3-0-120
Degree 22
Conductor 18151815
Sign 11
Analytic cond. 107.088107.088
Root an. cond. 10.348310.3483
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.34·2-s − 3·3-s + 20.6·4-s + 5·5-s − 16.0·6-s − 17.2·7-s + 67.4·8-s + 9·9-s + 26.7·10-s − 61.8·12-s + 8.03·13-s − 92.2·14-s − 15·15-s + 195.·16-s − 20.9·17-s + 48.1·18-s + 150.·19-s + 103.·20-s + 51.7·21-s + 150.·23-s − 202.·24-s + 25·25-s + 42.9·26-s − 27·27-s − 355.·28-s − 215.·29-s − 80.2·30-s + ⋯
L(s)  = 1  + 1.89·2-s − 0.577·3-s + 2.57·4-s + 0.447·5-s − 1.09·6-s − 0.931·7-s + 2.97·8-s + 0.333·9-s + 0.845·10-s − 1.48·12-s + 0.171·13-s − 1.76·14-s − 0.258·15-s + 3.05·16-s − 0.299·17-s + 0.630·18-s + 1.82·19-s + 1.15·20-s + 0.537·21-s + 1.36·23-s − 1.72·24-s + 0.200·25-s + 0.324·26-s − 0.192·27-s − 2.39·28-s − 1.37·29-s − 0.488·30-s + ⋯

Functional equation

Λ(s)=(1815s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1815s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18151815    =    3⋅5⋅1123 \cdot 5 \cdot 11^{2}
Sign: 11
Analytic conductor: 107.088107.088
Root analytic conductor: 10.348310.3483
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1815, ( :3/2), 1)(2,\ 1815,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 7.2492373437.249237343
L(12)L(\frac12) ≈\approx 7.2492373437.249237343
L(52)L(\frac{5}{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+3T 1 + 3T
5 1−5T 1 - 5T
11 1 1
good2 1−5.34T+8T2 1 - 5.34T + 8T^{2}
7 1+17.2T+343T2 1 + 17.2T + 343T^{2}
13 1−8.03T+2.19e3T2 1 - 8.03T + 2.19e3T^{2}
17 1+20.9T+4.91e3T2 1 + 20.9T + 4.91e3T^{2}
19 1−150.T+6.85e3T2 1 - 150.T + 6.85e3T^{2}
23 1−150.T+1.21e4T2 1 - 150.T + 1.21e4T^{2}
29 1+215.T+2.43e4T2 1 + 215.T + 2.43e4T^{2}
31 1+11.0T+2.97e4T2 1 + 11.0T + 2.97e4T^{2}
37 1+131.T+5.06e4T2 1 + 131.T + 5.06e4T^{2}
41 1−61.1T+6.89e4T2 1 - 61.1T + 6.89e4T^{2}
43 1−337.T+7.95e4T2 1 - 337.T + 7.95e4T^{2}
47 1+5.27T+1.03e5T2 1 + 5.27T + 1.03e5T^{2}
53 1−747.T+1.48e5T2 1 - 747.T + 1.48e5T^{2}
59 1+492.T+2.05e5T2 1 + 492.T + 2.05e5T^{2}
61 1−766.T+2.26e5T2 1 - 766.T + 2.26e5T^{2}
67 1−807.T+3.00e5T2 1 - 807.T + 3.00e5T^{2}
71 1+516.T+3.57e5T2 1 + 516.T + 3.57e5T^{2}
73 1−769.T+3.89e5T2 1 - 769.T + 3.89e5T^{2}
79 1−108.T+4.93e5T2 1 - 108.T + 4.93e5T^{2}
83 1−280.T+5.71e5T2 1 - 280.T + 5.71e5T^{2}
89 1−379.T+7.04e5T2 1 - 379.T + 7.04e5T^{2}
97 1+1.27e3T+9.12e5T2 1 + 1.27e3T + 9.12e5T^{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

−9.107647706506929907118635035245, −7.48910919666661823649337252826, −6.98342593883051615622079322266, −6.23937895647582191355515965912, −5.47805624185853025382838741469, −5.07927566413406014728237320668, −3.89011763407546398641749332785, −3.24555745355426819042546724602, −2.29832199180087388270507584165, −1.01312022908241089503859133362, 1.01312022908241089503859133362, 2.29832199180087388270507584165, 3.24555745355426819042546724602, 3.89011763407546398641749332785, 5.07927566413406014728237320668, 5.47805624185853025382838741469, 6.23937895647582191355515965912, 6.98342593883051615622079322266, 7.48910919666661823649337252826, 9.107647706506929907118635035245

Graph of the ZZ-function along the critical line