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Properties

Label 2-1260-1260.727-c0-0-1
Degree 22
Conductor 12601260
Sign 0.784+0.619i0.784 + 0.619i
Analytic cond. 0.6288210.628821
Root an. cond. 0.7929820.792982
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  + (−0.5 + 0.866i)2-s + (−0.707 + 0.707i)3-s + (−0.499 − 0.866i)4-s + (−0.258 + 0.965i)5-s + (−0.258 − 0.965i)6-s + (−0.965 + 0.258i)7-s + 0.999·8-s − 1.00i·9-s + (−0.707 − 0.707i)10-s + (−0.866 − 0.5i)11-s + (0.965 + 0.258i)12-s + (0.258 − 0.965i)13-s + (0.258 − 0.965i)14-s + (−0.500 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.707 + 0.707i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.707 + 0.707i)3-s + (−0.499 − 0.866i)4-s + (−0.258 + 0.965i)5-s + (−0.258 − 0.965i)6-s + (−0.965 + 0.258i)7-s + 0.999·8-s − 1.00i·9-s + (−0.707 − 0.707i)10-s + (−0.866 − 0.5i)11-s + (0.965 + 0.258i)12-s + (0.258 − 0.965i)13-s + (0.258 − 0.965i)14-s + (−0.500 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.707 + 0.707i)17-s + ⋯

Functional equation

Λ(s)=(1260s/2 ΓC(s) L(s)=((0.784+0.619i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1260s/2 ΓC(s) L(s)=((0.784+0.619i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12601260    =    22⋅32⋅5⋅72^{2} \cdot 3^{2} \cdot 5 \cdot 7
Sign: 0.784+0.619i0.784 + 0.619i
Analytic conductor: 0.6288210.628821
Root analytic conductor: 0.7929820.792982
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1260(727,⋅)\chi_{1260} (727, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1260, ( :0), 0.784+0.619i)(2,\ 1260,\ (\ :0),\ 0.784 + 0.619i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.16245056760.1624505676
L(12)L(\frac12) ≈\approx 0.16245056760.1624505676
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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
3 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
5 1+(0.258−0.965i)T 1 + (0.258 - 0.965i)T
7 1+(0.965−0.258i)T 1 + (0.965 - 0.258i)T
good11 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
13 1+(−0.258+0.965i)T+(−0.866−0.5i)T2 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2}
17 1+(0.707−0.707i)T−iT2 1 + (0.707 - 0.707i)T - iT^{2}
19 1+1.41iT−T2 1 + 1.41iT - T^{2}
23 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
29 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
31 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
37 1+iT2 1 + iT^{2}
41 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
43 1+(1.36−0.366i)T+(0.866−0.5i)T2 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2}
47 1+(−0.965+0.258i)T+(0.866−0.5i)T2 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2}
53 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(1.22+0.707i)T+(0.5+0.866i)T2 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2}
67 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
71 1−iT−T2 1 - iT - T^{2}
73 1+(0.707+0.707i)T+iT2 1 + (0.707 + 0.707i)T + iT^{2}
79 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
83 1+(0.258+0.965i)T+(−0.866+0.5i)T2 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2}
89 1+T2 1 + T^{2}
97 1+(−0.258−0.965i)T+(−0.866+0.5i)T2 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)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

−9.865285793930235054401279603003, −9.005880714344387332215923688746, −8.209774990372482785658539717299, −7.12783076252780754221279975980, −6.45651873690275570782728486654, −5.82906377687909698924198057211, −4.98971840061662865893144092170, −3.78631325972753904022059746087, −2.77688811522051189595600002347, −0.18737140778817125377059364956, 1.36653042965177314555976776395, 2.47909056263184480900536016581, 3.91498746518809432027181761464, 4.72399485465396908515286881164, 5.74746185732176705886632246841, 6.91141907078004746229230815986, 7.58406499049621960556002750981, 8.434818982429577261988082960410, 9.260336588053680347060852244084, 10.03917375717574547706833039787

Graph of the ZZ-function along the critical line