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Properties

Label 2-3381-1.1-c1-0-42
Degree 22
Conductor 33813381
Sign 11
Analytic cond. 26.997426.9974
Root an. cond. 5.195905.19590
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  − 0.329·2-s + 3-s − 1.89·4-s + 2.73·5-s − 0.329·6-s + 1.28·8-s + 9-s − 0.902·10-s − 2.50·11-s − 1.89·12-s − 1.48·13-s + 2.73·15-s + 3.35·16-s − 0.902·17-s − 0.329·18-s − 2.50·19-s − 5.17·20-s + 0.825·22-s + 23-s + 1.28·24-s + 2.48·25-s + 0.489·26-s + 27-s + 6.68·29-s − 0.902·30-s − 1.09·31-s − 3.67·32-s + ⋯
L(s)  = 1  − 0.233·2-s + 0.577·3-s − 0.945·4-s + 1.22·5-s − 0.134·6-s + 0.453·8-s + 0.333·9-s − 0.285·10-s − 0.755·11-s − 0.545·12-s − 0.411·13-s + 0.706·15-s + 0.839·16-s − 0.218·17-s − 0.0777·18-s − 0.574·19-s − 1.15·20-s + 0.176·22-s + 0.208·23-s + 0.261·24-s + 0.497·25-s + 0.0960·26-s + 0.192·27-s + 1.24·29-s − 0.164·30-s − 0.197·31-s − 0.649·32-s + ⋯

Functional equation

Λ(s)=(3381s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3381s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33813381    =    3⋅72⋅233 \cdot 7^{2} \cdot 23
Sign: 11
Analytic conductor: 26.997426.9974
Root analytic conductor: 5.195905.19590
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3381, ( :1/2), 1)(2,\ 3381,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9495830001.949583000
L(12)L(\frac12) ≈\approx 1.9495830001.949583000
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
7 1 1
23 1−T 1 - T
good2 1+0.329T+2T2 1 + 0.329T + 2T^{2}
5 1−2.73T+5T2 1 - 2.73T + 5T^{2}
11 1+2.50T+11T2 1 + 2.50T + 11T^{2}
13 1+1.48T+13T2 1 + 1.48T + 13T^{2}
17 1+0.902T+17T2 1 + 0.902T + 17T^{2}
19 1+2.50T+19T2 1 + 2.50T + 19T^{2}
29 1−6.68T+29T2 1 - 6.68T + 29T^{2}
31 1+1.09T+31T2 1 + 1.09T + 31T^{2}
37 1−9.91T+37T2 1 - 9.91T + 37T^{2}
41 1−2.30T+41T2 1 - 2.30T + 41T^{2}
43 1−5.83T+43T2 1 - 5.83T + 43T^{2}
47 1−6.74T+47T2 1 - 6.74T + 47T^{2}
53 1+4.91T+53T2 1 + 4.91T + 53T^{2}
59 1−7.32T+59T2 1 - 7.32T + 59T^{2}
61 1−1.00T+61T2 1 - 1.00T + 61T^{2}
67 1−11.2T+67T2 1 - 11.2T + 67T^{2}
71 1+0.362T+71T2 1 + 0.362T + 71T^{2}
73 1−5.34T+73T2 1 - 5.34T + 73T^{2}
79 1−10.4T+79T2 1 - 10.4T + 79T^{2}
83 1+7.59T+83T2 1 + 7.59T + 83T^{2}
89 1+10.6T+89T2 1 + 10.6T + 89T^{2}
97 1+13.9T+97T2 1 + 13.9T + 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

−8.648845447849085772181750684315, −8.088952948073867262480643504494, −7.28995337365300565086589262814, −6.28920396041634871778143616391, −5.53601479610653638086036111760, −4.78130119987258221374666823300, −4.04430546859387324448689388063, −2.80143040896507151707403441487, −2.13373336035446610503052003188, −0.852861991991762337128025979377, 0.852861991991762337128025979377, 2.13373336035446610503052003188, 2.80143040896507151707403441487, 4.04430546859387324448689388063, 4.78130119987258221374666823300, 5.53601479610653638086036111760, 6.28920396041634871778143616391, 7.28995337365300565086589262814, 8.088952948073867262480643504494, 8.648845447849085772181750684315

Graph of the ZZ-function along the critical line