↑

Properties

Label 2-1008-63.59-c1-0-44
Degree 22
Conductor 10081008
Sign −0.905−0.423i-0.905 - 0.423i
Analytic cond. 8.048928.04892
Root an. cond. 2.837062.83706
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.361 − 1.69i)3-s − 0.900·5-s + (−1.05 − 2.42i)7-s + (−2.73 + 1.22i)9-s − 3.12i·11-s + (1.99 + 1.14i)13-s + (0.325 + 1.52i)15-s + (2.57 − 4.46i)17-s + (−2.38 + 1.37i)19-s + (−3.72 + 2.66i)21-s + 1.71i·23-s − 4.18·25-s + (3.06 + 4.19i)27-s + (−1.85 + 1.07i)29-s + (−8.66 + 5.00i)31-s + ⋯
L(s)  = 1  + (−0.208 − 0.977i)3-s − 0.402·5-s + (−0.399 − 0.916i)7-s + (−0.912 + 0.408i)9-s − 0.943i·11-s + (0.552 + 0.318i)13-s + (0.0840 + 0.393i)15-s + (0.624 − 1.08i)17-s + (−0.546 + 0.315i)19-s + (−0.813 + 0.582i)21-s + 0.357i·23-s − 0.837·25-s + (0.589 + 0.807i)27-s + (−0.344 + 0.198i)29-s + (−1.55 + 0.898i)31-s + ⋯

Functional equation

Λ(s)=(1008s/2 ΓC(s) L(s)=((−0.905−0.423i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1008s/2 ΓC(s+1/2) L(s)=((−0.905−0.423i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10081008    =    24⋅32⋅72^{4} \cdot 3^{2} \cdot 7
Sign: −0.905−0.423i-0.905 - 0.423i
Analytic conductor: 8.048928.04892
Root analytic conductor: 2.837062.83706
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1008(689,⋅)\chi_{1008} (689, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1008, ( :1/2), −0.905−0.423i)(2,\ 1008,\ (\ :1/2),\ -0.905 - 0.423i)

Particular Values

L(1)L(1) ≈\approx 0.51538357330.5153835733
L(12)L(\frac12) ≈\approx 0.51538357330.5153835733
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)
bad2 1 1
3 1+(0.361+1.69i)T 1 + (0.361 + 1.69i)T
7 1+(1.05+2.42i)T 1 + (1.05 + 2.42i)T
good5 1+0.900T+5T2 1 + 0.900T + 5T^{2}
11 1+3.12iT−11T2 1 + 3.12iT - 11T^{2}
13 1+(−1.99−1.14i)T+(6.5+11.2i)T2 1 + (-1.99 - 1.14i)T + (6.5 + 11.2i)T^{2}
17 1+(−2.57+4.46i)T+(−8.5−14.7i)T2 1 + (-2.57 + 4.46i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.38−1.37i)T+(9.5−16.4i)T2 1 + (2.38 - 1.37i)T + (9.5 - 16.4i)T^{2}
23 1−1.71iT−23T2 1 - 1.71iT - 23T^{2}
29 1+(1.85−1.07i)T+(14.5−25.1i)T2 1 + (1.85 - 1.07i)T + (14.5 - 25.1i)T^{2}
31 1+(8.66−5.00i)T+(15.5−26.8i)T2 1 + (8.66 - 5.00i)T + (15.5 - 26.8i)T^{2}
37 1+(4.73+8.20i)T+(−18.5+32.0i)T2 1 + (4.73 + 8.20i)T + (-18.5 + 32.0i)T^{2}
41 1+(1.22−2.11i)T+(−20.5−35.5i)T2 1 + (1.22 - 2.11i)T + (-20.5 - 35.5i)T^{2}
43 1+(−0.273−0.473i)T+(−21.5+37.2i)T2 1 + (-0.273 - 0.473i)T + (-21.5 + 37.2i)T^{2}
47 1+(3.93−6.80i)T+(−23.5−40.7i)T2 1 + (3.93 - 6.80i)T + (-23.5 - 40.7i)T^{2}
53 1+(−12.0−6.97i)T+(26.5+45.8i)T2 1 + (-12.0 - 6.97i)T + (26.5 + 45.8i)T^{2}
59 1+(3.99+6.91i)T+(−29.5+51.0i)T2 1 + (3.99 + 6.91i)T + (-29.5 + 51.0i)T^{2}
61 1+(−6.28−3.62i)T+(30.5+52.8i)T2 1 + (-6.28 - 3.62i)T + (30.5 + 52.8i)T^{2}
67 1+(−1.83−3.17i)T+(−33.5+58.0i)T2 1 + (-1.83 - 3.17i)T + (-33.5 + 58.0i)T^{2}
71 1+14.1iT−71T2 1 + 14.1iT - 71T^{2}
73 1+(10.9+6.30i)T+(36.5+63.2i)T2 1 + (10.9 + 6.30i)T + (36.5 + 63.2i)T^{2}
79 1+(3.27−5.67i)T+(−39.5−68.4i)T2 1 + (3.27 - 5.67i)T + (-39.5 - 68.4i)T^{2}
83 1+(−0.184−0.319i)T+(−41.5+71.8i)T2 1 + (-0.184 - 0.319i)T + (-41.5 + 71.8i)T^{2}
89 1+(6.00+10.3i)T+(−44.5+77.0i)T2 1 + (6.00 + 10.3i)T + (-44.5 + 77.0i)T^{2}
97 1+(8.86−5.12i)T+(48.5−84.0i)T2 1 + (8.86 - 5.12i)T + (48.5 - 84.0i)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

−9.361878628526690691575509727989, −8.547907985556225255109058202091, −7.57977423379297751333823031328, −7.14263233392248998292521887672, −6.14190501449364163457916112533, −5.37339286353388065649732234214, −3.92721297073681447843942498742, −3.11831593765259594234852821768, −1.53610559016826828270135673626, −0.23933500260935710587043960458, 2.13118556786636357034697960012, 3.47062289842439872762669292604, 4.14736531046957261348150548124, 5.33617902262053264922019114746, 5.93375734454920076460044367318, 6.99461109955420921091335101068, 8.261939397379523378202859274011, 8.755022169346718595949994346832, 9.794712805095502600927746717001, 10.22675373540734404039090758433

Graph of the ZZ-function along the critical line