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Properties

Label 2-15e3-15.14-c0-0-5
Degree 22
Conductor 33753375
Sign 11
Analytic cond. 1.684341.68434
Root an. cond. 1.297821.29782
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.33·2-s + 0.790·4-s − 0.279·8-s − 1.16·16-s + 1.82·17-s + 1.33·19-s − 0.209·23-s + 1.82·31-s − 1.27·32-s + 2.44·34-s + 1.79·38-s − 0.279·46-s + 0.618·47-s + 49-s − 1.95·53-s − 1.95·61-s + 2.44·62-s − 0.547·64-s + 1.44·68-s + 1.05·76-s − 1.95·79-s − 1.95·83-s − 0.165·92-s + 0.827·94-s + 1.33·98-s − 2.61·106-s − 1.61·107-s + ⋯
L(s)  = 1  + 1.33·2-s + 0.790·4-s − 0.279·8-s − 1.16·16-s + 1.82·17-s + 1.33·19-s − 0.209·23-s + 1.82·31-s − 1.27·32-s + 2.44·34-s + 1.79·38-s − 0.279·46-s + 0.618·47-s + 49-s − 1.95·53-s − 1.95·61-s + 2.44·62-s − 0.547·64-s + 1.44·68-s + 1.05·76-s − 1.95·79-s − 1.95·83-s − 0.165·92-s + 0.827·94-s + 1.33·98-s − 2.61·106-s − 1.61·107-s + ⋯

Functional equation

Λ(s)=(3375s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(3375s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33753375    =    33⋅533^{3} \cdot 5^{3}
Sign: 11
Analytic conductor: 1.684341.68434
Root analytic conductor: 1.297821.29782
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3375(3374,⋅)\chi_{3375} (3374, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3375, ( :0), 1)(2,\ 3375,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.5384360452.538436045
L(12)L(\frac12) ≈\approx 2.5384360452.538436045
L(1)L(1) 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
5 1 1
good2 1−1.33T+T2 1 - 1.33T + T^{2}
7 1−T2 1 - T^{2}
11 1−T2 1 - T^{2}
13 1−T2 1 - T^{2}
17 1−1.82T+T2 1 - 1.82T + T^{2}
19 1−1.33T+T2 1 - 1.33T + T^{2}
23 1+0.209T+T2 1 + 0.209T + T^{2}
29 1−T2 1 - T^{2}
31 1−1.82T+T2 1 - 1.82T + T^{2}
37 1−T2 1 - T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1−0.618T+T2 1 - 0.618T + T^{2}
53 1+1.95T+T2 1 + 1.95T + T^{2}
59 1−T2 1 - T^{2}
61 1+1.95T+T2 1 + 1.95T + T^{2}
67 1−T2 1 - T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1+1.95T+T2 1 + 1.95T + T^{2}
83 1+1.95T+T2 1 + 1.95T + T^{2}
89 1−T2 1 - T^{2}
97 1−T2 1 - T^{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

−8.757559404778757287260412450887, −7.84074538217986916015680000977, −7.23053151582233900614204327311, −6.18778841810694993695568679747, −5.69712574898164366499901375615, −4.96009851236727662115523264904, −4.22696648210132150076729840492, −3.25941969567978494669220389188, −2.81707069808496930006828393459, −1.28555287507302653788962362903, 1.28555287507302653788962362903, 2.81707069808496930006828393459, 3.25941969567978494669220389188, 4.22696648210132150076729840492, 4.96009851236727662115523264904, 5.69712574898164366499901375615, 6.18778841810694993695568679747, 7.23053151582233900614204327311, 7.84074538217986916015680000977, 8.757559404778757287260412450887

Graph of the ZZ-function along the critical line