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Properties

Label 2-3872-8.5-c1-0-36
Degree 22
Conductor 38723872
Sign 0.925−0.379i0.925 - 0.379i
Analytic cond. 30.918030.9180
Root an. cond. 5.560405.56040
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.28i·3-s − 1.58i·5-s − 1.86·7-s + 1.34·9-s − 0.322i·13-s + 2.03·15-s − 4.79·17-s − 3.38i·19-s − 2.40i·21-s − 3.08·23-s + 2.49·25-s + 5.58i·27-s + 10.4i·29-s + 2.82·31-s + 2.95i·35-s + ⋯
L(s)  = 1  + 0.742i·3-s − 0.707i·5-s − 0.706·7-s + 0.448·9-s − 0.0893i·13-s + 0.525·15-s − 1.16·17-s − 0.776i·19-s − 0.524i·21-s − 0.642·23-s + 0.499·25-s + 1.07i·27-s + 1.93i·29-s + 0.506·31-s + 0.499i·35-s + ⋯

Functional equation

Λ(s)=(3872s/2 ΓC(s) L(s)=((0.925−0.379i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3872s/2 ΓC(s+1/2) L(s)=((0.925−0.379i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38723872    =    25⋅1122^{5} \cdot 11^{2}
Sign: 0.925−0.379i0.925 - 0.379i
Analytic conductor: 30.918030.9180
Root analytic conductor: 5.560405.56040
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3872(1937,⋅)\chi_{3872} (1937, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3872, ( :1/2), 0.925−0.379i)(2,\ 3872,\ (\ :1/2),\ 0.925 - 0.379i)

Particular Values

L(1)L(1) ≈\approx 1.6427056501.642705650
L(12)L(\frac12) ≈\approx 1.6427056501.642705650
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
11 1 1
good3 1−1.28iT−3T2 1 - 1.28iT - 3T^{2}
5 1+1.58iT−5T2 1 + 1.58iT - 5T^{2}
7 1+1.86T+7T2 1 + 1.86T + 7T^{2}
13 1+0.322iT−13T2 1 + 0.322iT - 13T^{2}
17 1+4.79T+17T2 1 + 4.79T + 17T^{2}
19 1+3.38iT−19T2 1 + 3.38iT - 19T^{2}
23 1+3.08T+23T2 1 + 3.08T + 23T^{2}
29 1−10.4iT−29T2 1 - 10.4iT - 29T^{2}
31 1−2.82T+31T2 1 - 2.82T + 31T^{2}
37 1−3.57iT−37T2 1 - 3.57iT - 37T^{2}
41 1−4.49T+41T2 1 - 4.49T + 41T^{2}
43 1+3.61iT−43T2 1 + 3.61iT - 43T^{2}
47 1−12.7T+47T2 1 - 12.7T + 47T^{2}
53 1+5.65iT−53T2 1 + 5.65iT - 53T^{2}
59 1−5.38iT−59T2 1 - 5.38iT - 59T^{2}
61 1+10.9iT−61T2 1 + 10.9iT - 61T^{2}
67 1+10.9iT−67T2 1 + 10.9iT - 67T^{2}
71 1−11.1T+71T2 1 - 11.1T + 71T^{2}
73 1−12.6T+73T2 1 - 12.6T + 73T^{2}
79 1−5.97T+79T2 1 - 5.97T + 79T^{2}
83 1+0.139iT−83T2 1 + 0.139iT - 83T^{2}
89 1−4.28T+89T2 1 - 4.28T + 89T^{2}
97 1−12.3T+97T2 1 - 12.3T + 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

−8.845801028306270590998803315897, −7.88091969288846463184995949223, −6.87810880287321255395343170959, −6.46313145948204869981284991432, −5.22761832873060003272979485431, −4.80300753719900881597025401565, −3.99213055519798057885344834664, −3.21135097150157044282960825818, −2.07971319911654571438348743164, −0.75285301824138727534315666219, 0.71150691335577342515415993470, 2.08866389139288901295802860397, 2.66406702943345438643949575406, 3.86506457504027702323870031237, 4.42685711645595034996344435904, 5.82636942114321960520203680230, 6.33545320982591088815334583596, 6.91195492894231923358089011517, 7.60045426301554049763447459207, 8.237292119557547954565521323114

Graph of the ZZ-function along the critical line