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Properties

Label 2-3528-168.131-c0-0-3
Degree 22
Conductor 35283528
Sign 0.686+0.726i0.686 + 0.726i
Analytic cond. 1.760701.76070
Root an. cond. 1.326911.32691
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.707 − 0.707i)8-s + (1.22 − 0.707i)11-s + (0.500 − 0.866i)16-s + (0.999 − i)22-s + (−0.707 + 1.22i)23-s + (−0.5 − 0.866i)25-s − 1.41·29-s + (0.258 − 0.965i)32-s + (1.73 + i)37-s + (0.707 − 1.22i)44-s + (−0.366 + 1.36i)46-s + (−0.707 − 0.707i)50-s + (−0.707 − 1.22i)53-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.707 − 0.707i)8-s + (1.22 − 0.707i)11-s + (0.500 − 0.866i)16-s + (0.999 − i)22-s + (−0.707 + 1.22i)23-s + (−0.5 − 0.866i)25-s − 1.41·29-s + (0.258 − 0.965i)32-s + (1.73 + i)37-s + (0.707 − 1.22i)44-s + (−0.366 + 1.36i)46-s + (−0.707 − 0.707i)50-s + (−0.707 − 1.22i)53-s + ⋯

Functional equation

Λ(s)=(3528s/2 ΓC(s) L(s)=((0.686+0.726i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3528s/2 ΓC(s) L(s)=((0.686+0.726i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35283528    =    23⋅32⋅722^{3} \cdot 3^{2} \cdot 7^{2}
Sign: 0.686+0.726i0.686 + 0.726i
Analytic conductor: 1.760701.76070
Root analytic conductor: 1.326911.32691
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3528(1979,⋅)\chi_{3528} (1979, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3528, ( :0), 0.686+0.726i)(2,\ 3528,\ (\ :0),\ 0.686 + 0.726i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.5241464042.524146404
L(12)L(\frac12) ≈\approx 2.5241464042.524146404
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+(−0.965+0.258i)T 1 + (-0.965 + 0.258i)T
3 1 1
7 1 1
good5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(−1.22+0.707i)T+(0.5−0.866i)T2 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2}
13 1+T2 1 + T^{2}
17 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(0.707−1.22i)T+(−0.5−0.866i)T2 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2}
29 1+1.41T+T2 1 + 1.41T + T^{2}
31 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
37 1+(−1.73−i)T+(0.5+0.866i)T2 1 + (-1.73 - i)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)T2 1 + (0.5 + 0.866i)T^{2}
53 1+(0.707+1.22i)T+(−0.5+0.866i)T2 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2}
59 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
61 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
67 1+(−1−1.73i)T+(−0.5+0.866i)T2 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2}
71 1+1.41T+T2 1 + 1.41T + T^{2}
73 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(−0.5−0.866i)T2 1 + (-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

−8.625854638277115596441935430656, −7.77005790090378364501770019708, −7.01436518272720185951271685555, −6.09154522849343054147690149676, −5.82980427558927010635687425734, −4.71603316994657986125154528973, −3.91707703809270213987051721592, −3.37037091654776793124497741623, −2.22464877370249704116077050575, −1.23423802939329208079921224439, 1.61020440235233608194342065241, 2.47615318281204067143685695725, 3.66324775930324827858933426828, 4.18827485866779442863563467666, 4.96082213196545948442205170682, 6.00477429332794889902822281522, 6.36530234104319437484615900489, 7.38296801220347390726632806891, 7.72786747434338625035652034460, 8.858539633106445500748875866384

Graph of the ZZ-function along the critical line