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Properties

Label 2-3864-3864.275-c0-0-0
Degree 22
Conductor 38643864
Sign −0.0633−0.997i-0.0633 - 0.997i
Analytic cond. 1.928381.92838
Root an. cond. 1.388661.38866
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s + (−0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + 0.999·14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.499 − 0.866i)18-s + (−0.499 + 0.866i)21-s − 0.999·22-s + (−0.5 + 0.866i)23-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s + (−0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + 0.999·14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.499 − 0.866i)18-s + (−0.499 + 0.866i)21-s − 0.999·22-s + (−0.5 + 0.866i)23-s + ⋯

Functional equation

Λ(s)=(3864s/2 ΓC(s) L(s)=((−0.0633−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3864s/2 ΓC(s) L(s)=((−0.0633−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38643864    =    23⋅3⋅7⋅232^{3} \cdot 3 \cdot 7 \cdot 23
Sign: −0.0633−0.997i-0.0633 - 0.997i
Analytic conductor: 1.928381.92838
Root analytic conductor: 1.388661.38866
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3864(275,⋅)\chi_{3864} (275, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3864, ( :0), −0.0633−0.997i)(2,\ 3864,\ (\ :0),\ -0.0633 - 0.997i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.49475941990.4947594199
L(12)L(\frac12) ≈\approx 0.49475941990.4947594199
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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
7 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
23 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
13 1−T2 1 - T^{2}
17 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
29 1+T+T2 1 + T + T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
53 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1+T+T2 1 + T + T^{2}
73 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
79 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
83 1−2T+T2 1 - 2T + T^{2}
89 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)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

−8.635164471155224299609633146046, −7.81883514142034311086079743865, −7.40467964676419264560926731991, −6.68828042689835933005012048544, −6.17193879448464788014814790169, −5.41192175869626677242436999120, −4.47959914242293995394590980714, −3.63311580300619851180057504939, −1.98430280576089188041396917564, −1.16918345609786118996748633236, 0.39728433309302425016243066789, 1.98439924577398963463871647379, 3.14460965188376305172179032263, 3.56565743517970142788511140951, 4.53439156329074768479390790282, 5.46825177460587599965927425734, 6.01626234519409887287532324077, 7.05797140902649174918237462276, 8.006432103935474332582634242024, 8.936458112476385394368884422230

Graph of the ZZ-function along the critical line