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Properties

Label 2-3724-1.1-c1-0-45
Degree 22
Conductor 37243724
Sign 11
Analytic cond. 29.736229.7362
Root an. cond. 5.453095.45309
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  + 3.23·3-s + 3.17·5-s + 7.48·9-s + 2.78·11-s − 5.61·13-s + 10.2·15-s + 2.92·17-s + 19-s + 6.95·23-s + 5.10·25-s + 14.5·27-s − 4.88·29-s − 4.91·31-s + 9.00·33-s + 1.15·37-s − 18.1·39-s − 4.32·41-s − 10.6·43-s + 23.7·45-s − 9.64·47-s + 9.45·51-s − 6.26·53-s + 8.83·55-s + 3.23·57-s − 5.91·59-s + 6.96·61-s − 17.8·65-s + ⋯
L(s)  = 1  + 1.86·3-s + 1.42·5-s + 2.49·9-s + 0.838·11-s − 1.55·13-s + 2.65·15-s + 0.708·17-s + 0.229·19-s + 1.45·23-s + 1.02·25-s + 2.79·27-s − 0.906·29-s − 0.883·31-s + 1.56·33-s + 0.190·37-s − 2.90·39-s − 0.674·41-s − 1.62·43-s + 3.54·45-s − 1.40·47-s + 1.32·51-s − 0.861·53-s + 1.19·55-s + 0.428·57-s − 0.769·59-s + 0.891·61-s − 2.21·65-s + ⋯

Functional equation

Λ(s)=(3724s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3724s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37243724    =    22⋅72⋅192^{2} \cdot 7^{2} \cdot 19
Sign: 11
Analytic conductor: 29.736229.7362
Root analytic conductor: 5.453095.45309
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3724, ( :1/2), 1)(2,\ 3724,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.2389834535.238983453
L(12)L(\frac12) ≈\approx 5.2389834535.238983453
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 1
7 1 1
19 1−T 1 - T
good3 1−3.23T+3T2 1 - 3.23T + 3T^{2}
5 1−3.17T+5T2 1 - 3.17T + 5T^{2}
11 1−2.78T+11T2 1 - 2.78T + 11T^{2}
13 1+5.61T+13T2 1 + 5.61T + 13T^{2}
17 1−2.92T+17T2 1 - 2.92T + 17T^{2}
23 1−6.95T+23T2 1 - 6.95T + 23T^{2}
29 1+4.88T+29T2 1 + 4.88T + 29T^{2}
31 1+4.91T+31T2 1 + 4.91T + 31T^{2}
37 1−1.15T+37T2 1 - 1.15T + 37T^{2}
41 1+4.32T+41T2 1 + 4.32T + 41T^{2}
43 1+10.6T+43T2 1 + 10.6T + 43T^{2}
47 1+9.64T+47T2 1 + 9.64T + 47T^{2}
53 1+6.26T+53T2 1 + 6.26T + 53T^{2}
59 1+5.91T+59T2 1 + 5.91T + 59T^{2}
61 1−6.96T+61T2 1 - 6.96T + 61T^{2}
67 1−0.156T+67T2 1 - 0.156T + 67T^{2}
71 1−1.41T+71T2 1 - 1.41T + 71T^{2}
73 1−11.8T+73T2 1 - 11.8T + 73T^{2}
79 1+10.8T+79T2 1 + 10.8T + 79T^{2}
83 1+3.48T+83T2 1 + 3.48T + 83T^{2}
89 1+7.79T+89T2 1 + 7.79T + 89T^{2}
97 1−18.2T+97T2 1 - 18.2T + 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

−8.715006386899096462249818364185, −7.82887823109003117612036425068, −7.13569964636920035186747439810, −6.58623164598755074111940001278, −5.35442409350998846362627690028, −4.72395929262658246535685385918, −3.52273313709299378174999463999, −2.95048219673612103731649088490, −2.02604116217972944272622510662, −1.47810733044619875012923818120, 1.47810733044619875012923818120, 2.02604116217972944272622510662, 2.95048219673612103731649088490, 3.52273313709299378174999463999, 4.72395929262658246535685385918, 5.35442409350998846362627690028, 6.58623164598755074111940001278, 7.13569964636920035186747439810, 7.82887823109003117612036425068, 8.715006386899096462249818364185

Graph of the ZZ-function along the critical line