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Properties

Label 2-2700-5.3-c0-0-3
Degree 22
Conductor 27002700
Sign 0.326+0.945i0.326 + 0.945i
Analytic cond. 1.347471.34747
Root an. cond. 1.160801.16080
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  + (−1.22 − 1.22i)13-s − 2i·19-s + 31-s + (1.22 − 1.22i)37-s + (−1.22 − 1.22i)43-s + i·49-s + 2·61-s + (−1.22 + 1.22i)67-s + (1.22 + 1.22i)73-s − i·79-s + (−1.22 − 1.22i)103-s − i·109-s + ⋯
L(s)  = 1  + (−1.22 − 1.22i)13-s − 2i·19-s + 31-s + (1.22 − 1.22i)37-s + (−1.22 − 1.22i)43-s + i·49-s + 2·61-s + (−1.22 + 1.22i)67-s + (1.22 + 1.22i)73-s − i·79-s + (−1.22 − 1.22i)103-s − i·109-s + ⋯

Functional equation

Λ(s)=(2700s/2 ΓC(s) L(s)=((0.326+0.945i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2700s/2 ΓC(s) L(s)=((0.326+0.945i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27002700    =    22⋅33⋅522^{2} \cdot 3^{3} \cdot 5^{2}
Sign: 0.326+0.945i0.326 + 0.945i
Analytic conductor: 1.347471.34747
Root analytic conductor: 1.160801.16080
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2700(2593,⋅)\chi_{2700} (2593, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2700, ( :0), 0.326+0.945i)(2,\ 2700,\ (\ :0),\ 0.326 + 0.945i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0226678981.022667898
L(12)L(\frac12) ≈\approx 1.0226678981.022667898
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)
bad2 1 1
3 1 1
5 1 1
good7 1−iT2 1 - iT^{2}
11 1+T2 1 + T^{2}
13 1+(1.22+1.22i)T+iT2 1 + (1.22 + 1.22i)T + iT^{2}
17 1−iT2 1 - iT^{2}
19 1+2iT−T2 1 + 2iT - T^{2}
23 1+iT2 1 + iT^{2}
29 1−T2 1 - T^{2}
31 1−T+T2 1 - T + T^{2}
37 1+(−1.22+1.22i)T−iT2 1 + (-1.22 + 1.22i)T - iT^{2}
41 1+T2 1 + T^{2}
43 1+(1.22+1.22i)T+iT2 1 + (1.22 + 1.22i)T + iT^{2}
47 1−iT2 1 - iT^{2}
53 1+iT2 1 + iT^{2}
59 1−T2 1 - T^{2}
61 1−2T+T2 1 - 2T + T^{2}
67 1+(1.22−1.22i)T−iT2 1 + (1.22 - 1.22i)T - iT^{2}
71 1+T2 1 + T^{2}
73 1+(−1.22−1.22i)T+iT2 1 + (-1.22 - 1.22i)T + iT^{2}
79 1+iT−T2 1 + iT - T^{2}
83 1+iT2 1 + iT^{2}
89 1−T2 1 - T^{2}
97 1−iT2 1 - iT^{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

−8.874755362378329211570210923988, −8.096285965999797326640339820756, −7.31452364580608769833795660672, −6.75143235765701948630419566653, −5.64764382446814432977078452388, −5.01640899342579018412891002441, −4.21237754841951144783603945009, −2.95367453565322765369292523219, −2.39064801727847079635113236791, −0.66704196694718698684290898265, 1.51530294258981332002446848611, 2.49280897995484442344852675242, 3.60906187662975308672006808065, 4.48273208693932452318536227393, 5.19374372976942847305689014475, 6.27949487807107560485534147151, 6.76013539513920542603117633402, 7.84175262947324996623184494635, 8.214001265338353712597708547347, 9.333291250172549011556694294631

Graph of the ZZ-function along the critical line