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Properties

Label 2-1800-40.13-c0-0-1
Degree 22
Conductor 18001800
Sign 0.229−0.973i0.229 - 0.973i
Analytic cond. 0.8983170.898317
Root an. cond. 0.9477950.947795
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (1 − i)7-s + (−0.707 + 0.707i)8-s + 1.41i·11-s + 1.41·14-s − 1.00·16-s + (−1.00 + 1.00i)22-s + (1.00 + 1.00i)28-s + 1.41·29-s + (−0.707 − 0.707i)32-s − 1.41·44-s − i·49-s + 1.41i·56-s + (1.00 + 1.00i)58-s − 1.41·59-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (1 − i)7-s + (−0.707 + 0.707i)8-s + 1.41i·11-s + 1.41·14-s − 1.00·16-s + (−1.00 + 1.00i)22-s + (1.00 + 1.00i)28-s + 1.41·29-s + (−0.707 − 0.707i)32-s − 1.41·44-s − i·49-s + 1.41i·56-s + (1.00 + 1.00i)58-s − 1.41·59-s + ⋯

Functional equation

Λ(s)=(1800s/2 ΓC(s) L(s)=((0.229−0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1800s/2 ΓC(s) L(s)=((0.229−0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18001800    =    23⋅32⋅522^{3} \cdot 3^{2} \cdot 5^{2}
Sign: 0.229−0.973i0.229 - 0.973i
Analytic conductor: 0.8983170.898317
Root analytic conductor: 0.9477950.947795
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1800(1693,⋅)\chi_{1800} (1693, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1800, ( :0), 0.229−0.973i)(2,\ 1800,\ (\ :0),\ 0.229 - 0.973i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7722654961.772265496
L(12)L(\frac12) ≈\approx 1.7722654961.772265496
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+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
3 1 1
5 1 1
good7 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
11 1−1.41iT−T2 1 - 1.41iT - T^{2}
13 1−iT2 1 - iT^{2}
17 1−iT2 1 - iT^{2}
19 1+T2 1 + T^{2}
23 1+iT2 1 + iT^{2}
29 1−1.41T+T2 1 - 1.41T + T^{2}
31 1+T2 1 + T^{2}
37 1+iT2 1 + iT^{2}
41 1+T2 1 + T^{2}
43 1−iT2 1 - iT^{2}
47 1−iT2 1 - iT^{2}
53 1−iT2 1 - iT^{2}
59 1+1.41T+T2 1 + 1.41T + T^{2}
61 1−T2 1 - T^{2}
67 1+iT2 1 + iT^{2}
71 1+T2 1 + T^{2}
73 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
79 1−T2 1 - T^{2}
83 1+(1.41−1.41i)T−iT2 1 + (1.41 - 1.41i)T - iT^{2}
89 1−T2 1 - T^{2}
97 1+(−1+i)T−iT2 1 + (-1 + i)T - 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

−9.590823972458598816218391845690, −8.539631843879044782965765458607, −7.83371924798380629531989275380, −7.22274378583398214906838951991, −6.60424128555511810470876836844, −5.47740960252324958243663073236, −4.49913603736955025405778657181, −4.33701070402582716012451235259, −2.97085972300348218395614389696, −1.69108136230916245936672200532, 1.25857026903807500460493415343, 2.45130827950648851578163056609, 3.23167243811718331006799878602, 4.36871636935297624993249702586, 5.19778009045072591400710342222, 5.83666076838975543748442152919, 6.58048612104266458918414747684, 7.929705517516424324645840620353, 8.655251282167984651836593106146, 9.238758406862070055022440777792

Graph of the ZZ-function along the critical line