↑

Properties

Label 2-1472-1472.413-c0-0-1
Degree 22
Conductor 14721472
Sign 0.881+0.471i0.881 + 0.471i
Analytic cond. 0.7346230.734623
Root an. cond. 0.8571010.857101
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.382 + 0.923i)2-s + (−0.382 − 0.0761i)3-s + (−0.707 − 0.707i)4-s + (0.216 − 0.324i)6-s + (0.923 − 0.382i)8-s + (−0.783 − 0.324i)9-s + (0.216 + 0.324i)12-s + (−0.216 − 0.324i)13-s + i·16-s + (0.599 − 0.599i)18-s + (0.382 − 0.923i)23-s + (−0.382 + 0.0761i)24-s + (−0.382 − 0.923i)25-s + (0.382 − 0.0761i)26-s + (0.599 + 0.400i)27-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)2-s + (−0.382 − 0.0761i)3-s + (−0.707 − 0.707i)4-s + (0.216 − 0.324i)6-s + (0.923 − 0.382i)8-s + (−0.783 − 0.324i)9-s + (0.216 + 0.324i)12-s + (−0.216 − 0.324i)13-s + i·16-s + (0.599 − 0.599i)18-s + (0.382 − 0.923i)23-s + (−0.382 + 0.0761i)24-s + (−0.382 − 0.923i)25-s + (0.382 − 0.0761i)26-s + (0.599 + 0.400i)27-s + ⋯

Functional equation

Λ(s)=(1472s/2 ΓC(s) L(s)=((0.881+0.471i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1472s/2 ΓC(s) L(s)=((0.881+0.471i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14721472    =    26⋅232^{6} \cdot 23
Sign: 0.881+0.471i0.881 + 0.471i
Analytic conductor: 0.7346230.734623
Root analytic conductor: 0.8571010.857101
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1472(413,⋅)\chi_{1472} (413, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1472, ( :0), 0.881+0.471i)(2,\ 1472,\ (\ :0),\ 0.881 + 0.471i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.56494185830.5649418583
L(12)L(\frac12) ≈\approx 0.56494185830.5649418583
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+(0.382−0.923i)T 1 + (0.382 - 0.923i)T
23 1+(−0.382+0.923i)T 1 + (-0.382 + 0.923i)T
good3 1+(0.382+0.0761i)T+(0.923+0.382i)T2 1 + (0.382 + 0.0761i)T + (0.923 + 0.382i)T^{2}
5 1+(0.382+0.923i)T2 1 + (0.382 + 0.923i)T^{2}
7 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
11 1+(0.923−0.382i)T2 1 + (0.923 - 0.382i)T^{2}
13 1+(0.216+0.324i)T+(−0.382+0.923i)T2 1 + (0.216 + 0.324i)T + (-0.382 + 0.923i)T^{2}
17 1−iT2 1 - iT^{2}
19 1+(−0.382+0.923i)T2 1 + (-0.382 + 0.923i)T^{2}
29 1+(−0.324+1.63i)T+(−0.923−0.382i)T2 1 + (-0.324 + 1.63i)T + (-0.923 - 0.382i)T^{2}
31 1−0.765iT−T2 1 - 0.765iT - T^{2}
37 1+(−0.382−0.923i)T2 1 + (-0.382 - 0.923i)T^{2}
41 1+(−0.765+1.84i)T+(−0.707−0.707i)T2 1 + (-0.765 + 1.84i)T + (-0.707 - 0.707i)T^{2}
43 1+(−0.923+0.382i)T2 1 + (-0.923 + 0.382i)T^{2}
47 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
53 1+(0.923−0.382i)T2 1 + (0.923 - 0.382i)T^{2}
59 1+(1.08−1.63i)T+(−0.382−0.923i)T2 1 + (1.08 - 1.63i)T + (-0.382 - 0.923i)T^{2}
61 1+(−0.923−0.382i)T2 1 + (-0.923 - 0.382i)T^{2}
67 1+(−0.923−0.382i)T2 1 + (-0.923 - 0.382i)T^{2}
71 1+(−1.70+0.707i)T+(0.707−0.707i)T2 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2}
73 1+(−1.30−0.541i)T+(0.707+0.707i)T2 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2}
79 1+iT2 1 + iT^{2}
83 1+(−0.382+0.923i)T2 1 + (-0.382 + 0.923i)T^{2}
89 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
97 1+T2 1 + T^{2}
show more
show less
   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

−9.502773636445319094921367300633, −8.636184898156146022643568278194, −8.133821140127717741130988127969, −7.12420939440511903529401135306, −6.38031355919030160921873671676, −5.70769083875309325573594794541, −4.89940591060155826179476359165, −3.87651153937540560657556111275, −2.41518409875495160665413968061, −0.58323883456524963885411174514, 1.41711752381582801047826787520, 2.68797926960873314046303226804, 3.54444734333588368451289385143, 4.72488352614317995457831663074, 5.38625195644639512617372997267, 6.52527687308122578185593793519, 7.64264638962786805602653356793, 8.226173239665050781684233710546, 9.324167336161224357393237010030, 9.582006840586143901858827412337

Graph of the ZZ-function along the critical line