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Properties

Label 2-1872-208.51-c0-0-2
Degree 22
Conductor 18721872
Sign 0.923+0.382i0.923 + 0.382i
Analytic cond. 0.9342490.934249
Root an. cond. 0.9665650.966565
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  + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (0.541 + 0.541i)5-s + (0.382 − 0.923i)8-s + (0.707 + 0.292i)10-s + (−0.541 + 0.541i)11-s + (0.707 + 0.707i)13-s − i·16-s + 0.765·20-s + (−0.292 + 0.707i)22-s − 0.414i·25-s + (0.923 + 0.382i)26-s + (−0.382 − 0.923i)32-s + (0.707 − 0.292i)40-s − 0.765·41-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (0.541 + 0.541i)5-s + (0.382 − 0.923i)8-s + (0.707 + 0.292i)10-s + (−0.541 + 0.541i)11-s + (0.707 + 0.707i)13-s − i·16-s + 0.765·20-s + (−0.292 + 0.707i)22-s − 0.414i·25-s + (0.923 + 0.382i)26-s + (−0.382 − 0.923i)32-s + (0.707 − 0.292i)40-s − 0.765·41-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=((0.923+0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s) L(s)=((0.923+0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: 0.923+0.382i0.923 + 0.382i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1872(883,⋅)\chi_{1872} (883, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1872, ( :0), 0.923+0.382i)(2,\ 1872,\ (\ :0),\ 0.923 + 0.382i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.1778167482.177816748
L(12)L(\frac12) ≈\approx 2.1778167482.177816748
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+(−0.923+0.382i)T 1 + (-0.923 + 0.382i)T
3 1 1
13 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
good5 1+(−0.541−0.541i)T+iT2 1 + (-0.541 - 0.541i)T + iT^{2}
7 1−T2 1 - T^{2}
11 1+(0.541−0.541i)T−iT2 1 + (0.541 - 0.541i)T - iT^{2}
17 1+T2 1 + T^{2}
19 1+iT2 1 + iT^{2}
23 1+T2 1 + T^{2}
29 1+iT2 1 + iT^{2}
31 1+T2 1 + T^{2}
37 1+iT2 1 + iT^{2}
41 1+0.765T+T2 1 + 0.765T + T^{2}
43 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
47 1+0.765T+T2 1 + 0.765T + T^{2}
53 1−iT2 1 - iT^{2}
59 1+(1.30−1.30i)T−iT2 1 + (1.30 - 1.30i)T - iT^{2}
61 1+iT2 1 + iT^{2}
67 1+iT2 1 + iT^{2}
71 1+0.765iT−T2 1 + 0.765iT - T^{2}
73 1+T2 1 + T^{2}
79 1+1.41iT−T2 1 + 1.41iT - T^{2}
83 1+(−1.30−1.30i)T+iT2 1 + (-1.30 - 1.30i)T + iT^{2}
89 1+1.84T+T2 1 + 1.84T + 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

−9.636087941770067442913858607211, −8.687697273127681205114119419709, −7.54614478949998433734026428295, −6.73679523140277139872964503385, −6.16070767542309768593672075791, −5.27235756799664577362089609121, −4.42957884129729841586529078190, −3.48747140725327233743232613661, −2.50439798157545175101046151325, −1.64845607135786557056170392997, 1.56416235192031470248687706679, 2.85991859565469638922578385891, 3.63596700927309134359909146750, 4.78987707332713478440354933360, 5.44376736328926327040473540444, 6.06899684750243352898975042395, 6.91438459423210596832551377373, 7.980166874251283940163855866120, 8.406070256309084620478632222397, 9.376114407207631243297890298767

Graph of the ZZ-function along the critical line