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Properties

Label 2-261-1.1-c1-0-5
Degree 22
Conductor 261261
Sign 11
Analytic cond. 2.084092.08409
Root an. cond. 1.443631.44363
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  + 1.93·2-s + 1.74·4-s − 0.508·5-s + 3.68·7-s − 0.491·8-s − 0.983·10-s + 0.318·11-s + 4.18·13-s + 7.12·14-s − 4.44·16-s − 3.17·17-s − 5.87·19-s − 0.887·20-s + 0.616·22-s − 2.50·23-s − 4.74·25-s + 8.10·26-s + 6.42·28-s − 29-s + 2.50·31-s − 7.61·32-s − 6.14·34-s − 1.87·35-s + 7.87·37-s − 11.3·38-s + 0.249·40-s − 8.72·41-s + ⋯
L(s)  = 1  + 1.36·2-s + 0.872·4-s − 0.227·5-s + 1.39·7-s − 0.173·8-s − 0.311·10-s + 0.0960·11-s + 1.16·13-s + 1.90·14-s − 1.11·16-s − 0.769·17-s − 1.34·19-s − 0.198·20-s + 0.131·22-s − 0.522·23-s − 0.948·25-s + 1.59·26-s + 1.21·28-s − 0.185·29-s + 0.450·31-s − 1.34·32-s − 1.05·34-s − 0.316·35-s + 1.29·37-s − 1.84·38-s + 0.0395·40-s − 1.36·41-s + ⋯

Functional equation

Λ(s)=(261s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(261s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 261261    =    32⋅293^{2} \cdot 29
Sign: 11
Analytic conductor: 2.084092.08409
Root analytic conductor: 1.443631.44363
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 261, ( :1/2), 1)(2,\ 261,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4667567772.466756777
L(12)L(\frac12) ≈\approx 2.4667567772.466756777
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 1
29 1+T 1 + T
good2 1−1.93T+2T2 1 - 1.93T + 2T^{2}
5 1+0.508T+5T2 1 + 0.508T + 5T^{2}
7 1−3.68T+7T2 1 - 3.68T + 7T^{2}
11 1−0.318T+11T2 1 - 0.318T + 11T^{2}
13 1−4.18T+13T2 1 - 4.18T + 13T^{2}
17 1+3.17T+17T2 1 + 3.17T + 17T^{2}
19 1+5.87T+19T2 1 + 5.87T + 19T^{2}
23 1+2.50T+23T2 1 + 2.50T + 23T^{2}
31 1−2.50T+31T2 1 - 2.50T + 31T^{2}
37 1−7.87T+37T2 1 - 7.87T + 37T^{2}
41 1+8.72T+41T2 1 + 8.72T + 41T^{2}
43 1+10.7T+43T2 1 + 10.7T + 43T^{2}
47 1−11.0T+47T2 1 - 11.0T + 47T^{2}
53 1−8.24T+53T2 1 - 8.24T + 53T^{2}
59 1−11.3T+59T2 1 - 11.3T + 59T^{2}
61 1+3.87T+61T2 1 + 3.87T + 61T^{2}
67 1−7.04T+67T2 1 - 7.04T + 67T^{2}
71 1+6.24T+71T2 1 + 6.24T + 71T^{2}
73 1+7.87T+73T2 1 + 7.87T + 73T^{2}
79 1+4.85T+79T2 1 + 4.85T + 79T^{2}
83 1−8.37T+83T2 1 - 8.37T + 83T^{2}
89 1−15.9T+89T2 1 - 15.9T + 89T^{2}
97 1−11.2T+97T2 1 - 11.2T + 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

−11.89379906390817673898495680908, −11.45717177468320183508910298287, −10.50317658005145421373983048345, −8.827804660270956228612202293003, −8.110151513408090971118370390688, −6.64955494139607739085014912079, −5.69070465123847630004924937603, −4.54613745325072059167389947720, −3.85551656049954225721533006205, −2.09940579716822489259011887500, 2.09940579716822489259011887500, 3.85551656049954225721533006205, 4.54613745325072059167389947720, 5.69070465123847630004924937603, 6.64955494139607739085014912079, 8.110151513408090971118370390688, 8.827804660270956228612202293003, 10.50317658005145421373983048345, 11.45717177468320183508910298287, 11.89379906390817673898495680908

Graph of the ZZ-function along the critical line