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Properties

Label 2-675-5.2-c0-0-1
Degree 22
Conductor 675675
Sign 0.945+0.326i0.945 + 0.326i
Analytic cond. 0.3368680.336868
Root an. cond. 0.5804040.580404
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  − i·4-s + (1.22 + 1.22i)7-s − 16-s − i·19-s + (1.22 − 1.22i)28-s + 31-s + (−1.22 − 1.22i)37-s + (−1.22 + 1.22i)43-s + 1.99i·49-s − 61-s + i·64-s + (−1.22 + 1.22i)73-s − 76-s + i·79-s + (−1.22 − 1.22i)97-s + ⋯
L(s)  = 1  − i·4-s + (1.22 + 1.22i)7-s − 16-s − i·19-s + (1.22 − 1.22i)28-s + 31-s + (−1.22 − 1.22i)37-s + (−1.22 + 1.22i)43-s + 1.99i·49-s − 61-s + i·64-s + (−1.22 + 1.22i)73-s − 76-s + i·79-s + (−1.22 − 1.22i)97-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=((0.945+0.326i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s) L(s)=((0.945+0.326i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: 0.945+0.326i0.945 + 0.326i
Analytic conductor: 0.3368680.336868
Root analytic conductor: 0.5804040.580404
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ675(82,⋅)\chi_{675} (82, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 675, ( :0), 0.945+0.326i)(2,\ 675,\ (\ :0),\ 0.945 + 0.326i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0349186851.034918685
L(12)L(\frac12) ≈\approx 1.0349186851.034918685
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)
bad3 1 1
5 1 1
good2 1+iT2 1 + iT^{2}
7 1+(−1.22−1.22i)T+iT2 1 + (-1.22 - 1.22i)T + iT^{2}
11 1+T2 1 + T^{2}
13 1−iT2 1 - iT^{2}
17 1+iT2 1 + iT^{2}
19 1+iT−T2 1 + iT - 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+T+T2 1 + T + T^{2}
67 1+iT2 1 + 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+(1.22+1.22i)T+iT2 1 + (1.22 + 1.22i)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

−10.78071287903503856332010540663, −9.762127944616193245298331333996, −8.924231481404348483644161427424, −8.285837107122008038410596797855, −7.06145238520667371942355739240, −6.00988172258741036538284739376, −5.22059484773257747419478587281, −4.54166633603933565975712175367, −2.66337317044612063450904882123, −1.59296781081227783987814082373, 1.68777958372204334564742477948, 3.30526644552585761737677076977, 4.22513977990903072098060814147, 5.03736764589016524922479413453, 6.55810132509333050460831109417, 7.45057385091042767982749692910, 8.066018408074557773732614864473, 8.740921047563922360928856367337, 10.10456985609110112495793942794, 10.72382500755232839908505086296

Graph of the ZZ-function along the critical line