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Properties

Label 2-3267-11.10-c0-0-3
Degree 22
Conductor 32673267
Sign 0.522+0.852i0.522 + 0.852i
Analytic cond. 1.630441.63044
Root an. cond. 1.276881.27688
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 0.517i·7-s − 1.93i·13-s + 16-s − 1.41i·19-s − 25-s − 0.517i·28-s − 1.73·31-s + 1.41i·43-s + 0.732·49-s − 1.93i·52-s + 1.41i·61-s + 64-s + 1.73·67-s − 1.93i·73-s + ⋯
L(s)  = 1  + 4-s − 0.517i·7-s − 1.93i·13-s + 16-s − 1.41i·19-s − 25-s − 0.517i·28-s − 1.73·31-s + 1.41i·43-s + 0.732·49-s − 1.93i·52-s + 1.41i·61-s + 64-s + 1.73·67-s − 1.93i·73-s + ⋯

Functional equation

Λ(s)=(3267s/2 ΓC(s) L(s)=((0.522+0.852i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3267s/2 ΓC(s) L(s)=((0.522+0.852i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32673267    =    33⋅1123^{3} \cdot 11^{2}
Sign: 0.522+0.852i0.522 + 0.852i
Analytic conductor: 1.630441.63044
Root analytic conductor: 1.276881.27688
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3267(2782,⋅)\chi_{3267} (2782, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3267, ( :0), 0.522+0.852i)(2,\ 3267,\ (\ :0),\ 0.522 + 0.852i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5369862321.536986232
L(12)L(\frac12) ≈\approx 1.5369862321.536986232
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
11 1 1
good2 1−T2 1 - T^{2}
5 1+T2 1 + T^{2}
7 1+0.517iT−T2 1 + 0.517iT - T^{2}
13 1+1.93iT−T2 1 + 1.93iT - T^{2}
17 1−T2 1 - T^{2}
19 1+1.41iT−T2 1 + 1.41iT - T^{2}
23 1+T2 1 + T^{2}
29 1−T2 1 - T^{2}
31 1+1.73T+T2 1 + 1.73T + T^{2}
37 1+T2 1 + T^{2}
41 1−T2 1 - T^{2}
43 1−1.41iT−T2 1 - 1.41iT - T^{2}
47 1+T2 1 + T^{2}
53 1+T2 1 + T^{2}
59 1+T2 1 + T^{2}
61 1−1.41iT−T2 1 - 1.41iT - T^{2}
67 1−1.73T+T2 1 - 1.73T + T^{2}
71 1+T2 1 + T^{2}
73 1+1.93iT−T2 1 + 1.93iT - T^{2}
79 1−1.93iT−T2 1 - 1.93iT - T^{2}
83 1−T2 1 - T^{2}
89 1+T2 1 + T^{2}
97 1−T+T2 1 - T + 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

−8.581398774273914779198166497835, −7.66212898031622835293350544831, −7.42491528764338064156200076446, −6.48615943713975337362666996633, −5.71557868154761541621362927897, −5.05556988636564458177397510377, −3.82070752023145197271420709173, −3.05550009227584157588296114351, −2.22634970037347322430525235748, −0.895633476524245239422701637981, 1.80857182052585382954070619772, 2.08904261085142173650190533901, 3.48356403681026982115537413227, 4.10294859997713759860091138422, 5.38588173126502152311447486104, 5.96527130959385211439344471664, 6.75574595240329875711417235166, 7.33473840741194380848079114498, 8.148198566491282453877487706603, 8.972913020748751643222746122869

Graph of the ZZ-function along the critical line