↑

Properties

Label 2-3267-11.8-c0-0-1
Degree 22
Conductor 32673267
Sign 0.776+0.629i0.776 + 0.629i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)4-s + (1.83 + 0.596i)7-s + (−0.304 − 0.418i)13-s + (−0.809 − 0.587i)16-s + (1.34 − 0.437i)19-s + (−0.309 − 0.951i)25-s + (1.13 − 1.56i)28-s + (−1.40 + 1.01i)31-s + 1.41i·43-s + (2.21 + 1.60i)49-s + (−0.492 + 0.159i)52-s + (0.831 − 1.14i)61-s + (−0.809 + 0.587i)64-s − 1.73·67-s + (0.492 + 0.159i)73-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)4-s + (1.83 + 0.596i)7-s + (−0.304 − 0.418i)13-s + (−0.809 − 0.587i)16-s + (1.34 − 0.437i)19-s + (−0.309 − 0.951i)25-s + (1.13 − 1.56i)28-s + (−1.40 + 1.01i)31-s + 1.41i·43-s + (2.21 + 1.60i)49-s + (−0.492 + 0.159i)52-s + (0.831 − 1.14i)61-s + (−0.809 + 0.587i)64-s − 1.73·67-s + (0.492 + 0.159i)73-s + ⋯

Functional equation

Λ(s)=(3267s/2 ΓC(s) L(s)=((0.776+0.629i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3267s/2 ΓC(s) L(s)=((0.776+0.629i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32673267    =    33⋅1123^{3} \cdot 11^{2}
Sign: 0.776+0.629i0.776 + 0.629i
Analytic conductor: 1.630441.63044
Root analytic conductor: 1.276881.27688
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3267(2296,⋅)\chi_{3267} (2296, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3267, ( :0), 0.776+0.629i)(2,\ 3267,\ (\ :0),\ 0.776 + 0.629i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6704548021.670454802
L(12)L(\frac12) ≈\approx 1.6704548021.670454802
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+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
5 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
7 1+(−1.83−0.596i)T+(0.809+0.587i)T2 1 + (-1.83 - 0.596i)T + (0.809 + 0.587i)T^{2}
13 1+(0.304+0.418i)T+(−0.309+0.951i)T2 1 + (0.304 + 0.418i)T + (-0.309 + 0.951i)T^{2}
17 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
19 1+(−1.34+0.437i)T+(0.809−0.587i)T2 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2}
23 1+T2 1 + T^{2}
29 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
31 1+(1.40−1.01i)T+(0.309−0.951i)T2 1 + (1.40 - 1.01i)T + (0.309 - 0.951i)T^{2}
37 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
41 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
43 1−1.41iT−T2 1 - 1.41iT - T^{2}
47 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
53 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
59 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
61 1+(−0.831+1.14i)T+(−0.309−0.951i)T2 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2}
67 1+1.73T+T2 1 + 1.73T + T^{2}
71 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
73 1+(−0.492−0.159i)T+(0.809+0.587i)T2 1 + (-0.492 - 0.159i)T + (0.809 + 0.587i)T^{2}
79 1+(−0.304−0.418i)T+(−0.309+0.951i)T2 1 + (-0.304 - 0.418i)T + (-0.309 + 0.951i)T^{2}
83 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
89 1+T2 1 + T^{2}
97 1+(0.809−0.587i)T+(0.309−0.951i)T2 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2}
show more
show less
   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

−8.778283416708278123001854072352, −7.923399406818103638840591730883, −7.39307124777618580017454041053, −6.41782736838824107089606649835, −5.40635447902892503915551444764, −5.19495858522531893981043954340, −4.36961790662430934089483371882, −2.92508562816897517109392922132, −1.99435865501308734993850452317, −1.18920143630124494389179214403, 1.46741829347561648117891561707, 2.26815971663040867106034850829, 3.54630121080593589671291324978, 4.14883819720093513771396099831, 5.03734489667721323162576234370, 5.73923556404982947143949327324, 7.16001962700275869855861589362, 7.39805918383502590889406739924, 7.990087919011386982709375293523, 8.760863659997982818502349970741

Graph of the ZZ-function along the critical line