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Properties

Label 2-675-45.4-c1-0-8
Degree 22
Conductor 675675
Sign 0.993−0.114i0.993 - 0.114i
Analytic cond. 5.389905.38990
Root an. cond. 2.321612.32161
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.500 + 0.866i)4-s + (2.59 − 1.5i)7-s + 3i·8-s + (−1 − 1.73i)11-s + (1.73 + i)13-s + (1.5 − 2.59i)14-s + (0.500 + 0.866i)16-s + 4i·17-s + 8·19-s + (−1.73 − 0.999i)22-s + (2.59 + 1.5i)23-s + 1.99·26-s + 3i·28-s + (0.5 + 0.866i)29-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.250 + 0.433i)4-s + (0.981 − 0.566i)7-s + 1.06i·8-s + (−0.301 − 0.522i)11-s + (0.480 + 0.277i)13-s + (0.400 − 0.694i)14-s + (0.125 + 0.216i)16-s + 0.970i·17-s + 1.83·19-s + (−0.369 − 0.213i)22-s + (0.541 + 0.312i)23-s + 0.392·26-s + 0.566i·28-s + (0.0928 + 0.160i)29-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=((0.993−0.114i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+1/2) L(s)=((0.993−0.114i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: 0.993−0.114i0.993 - 0.114i
Analytic conductor: 5.389905.38990
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ675(199,⋅)\chi_{675} (199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 675, ( :1/2), 0.993−0.114i)(2,\ 675,\ (\ :1/2),\ 0.993 - 0.114i)

Particular Values

L(1)L(1) ≈\approx 2.14741+0.123162i2.14741 + 0.123162i
L(12)L(\frac12) ≈\approx 2.14741+0.123162i2.14741 + 0.123162i
L(32)L(\frac{3}{2}) 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+(−0.866+0.5i)T+(1−1.73i)T2 1 + (-0.866 + 0.5i)T + (1 - 1.73i)T^{2}
7 1+(−2.59+1.5i)T+(3.5−6.06i)T2 1 + (-2.59 + 1.5i)T + (3.5 - 6.06i)T^{2}
11 1+(1+1.73i)T+(−5.5+9.52i)T2 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2}
13 1+(−1.73−i)T+(6.5+11.2i)T2 1 + (-1.73 - i)T + (6.5 + 11.2i)T^{2}
17 1−4iT−17T2 1 - 4iT - 17T^{2}
19 1−8T+19T2 1 - 8T + 19T^{2}
23 1+(−2.59−1.5i)T+(11.5+19.9i)T2 1 + (-2.59 - 1.5i)T + (11.5 + 19.9i)T^{2}
29 1+(−0.5−0.866i)T+(−14.5+25.1i)T2 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2}
31 1+(−15.5−26.8i)T2 1 + (-15.5 - 26.8i)T^{2}
37 1−4iT−37T2 1 - 4iT - 37T^{2}
41 1+(−2.5+4.33i)T+(−20.5−35.5i)T2 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2}
43 1+(6.92−4i)T+(21.5−37.2i)T2 1 + (6.92 - 4i)T + (21.5 - 37.2i)T^{2}
47 1+(−6.06+3.5i)T+(23.5−40.7i)T2 1 + (-6.06 + 3.5i)T + (23.5 - 40.7i)T^{2}
53 1−2iT−53T2 1 - 2iT - 53T^{2}
59 1+(−7+12.1i)T+(−29.5−51.0i)T2 1 + (-7 + 12.1i)T + (-29.5 - 51.0i)T^{2}
61 1+(3.5+6.06i)T+(−30.5+52.8i)T2 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2}
67 1+(2.59+1.5i)T+(33.5+58.0i)T2 1 + (2.59 + 1.5i)T + (33.5 + 58.0i)T^{2}
71 1+2T+71T2 1 + 2T + 71T^{2}
73 1−4iT−73T2 1 - 4iT - 73T^{2}
79 1+(3+5.19i)T+(−39.5+68.4i)T2 1 + (3 + 5.19i)T + (-39.5 + 68.4i)T^{2}
83 1+(7.79−4.5i)T+(41.5−71.8i)T2 1 + (7.79 - 4.5i)T + (41.5 - 71.8i)T^{2}
89 1+15T+89T2 1 + 15T + 89T^{2}
97 1+(1.73−i)T+(48.5−84.0i)T2 1 + (1.73 - i)T + (48.5 - 84.0i)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

−10.90642698527740960415877920159, −9.713116724960405718340243398995, −8.572068195593347711592871001881, −8.019724471807744644162704460656, −7.12338386369203983760094452322, −5.68368825407470785520033138161, −4.91424739737909736192938338204, −3.91994113874144726662661087229, −3.05509898669358670064665512742, −1.47004028892139664765682362741, 1.20513933024391575085177907668, 2.84034043073368269531378945440, 4.25138102848267378950118637080, 5.21228152956657950829616000738, 5.59384290486228412745919484861, 6.92292644948889585028588805074, 7.69455030349641775089774965689, 8.823099313681104013352343971600, 9.586403667212809748677897649958, 10.45420552087374898278983221479

Graph of the ZZ-function along the critical line