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Properties

Label 2-768-16.13-c1-0-14
Degree 22
Conductor 768768
Sign −0.382+0.923i-0.382 + 0.923i
Analytic cond. 6.132516.13251
Root an. cond. 2.476392.47639
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  + (−0.707 − 0.707i)3-s + (2 − 2i)5-s − 4.24i·7-s + 1.00i·9-s + (2.82 − 2.82i)11-s + (3 + 3i)13-s − 2.82·15-s − 6·17-s + (−1.41 − 1.41i)19-s + (−3 + 3i)21-s + 2.82i·23-s − 3i·25-s + (0.707 − 0.707i)27-s + (4 + 4i)29-s − 4.24·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.894 − 0.894i)5-s − 1.60i·7-s + 0.333i·9-s + (0.852 − 0.852i)11-s + (0.832 + 0.832i)13-s − 0.730·15-s − 1.45·17-s + (−0.324 − 0.324i)19-s + (−0.654 + 0.654i)21-s + 0.589i·23-s − 0.600i·25-s + (0.136 − 0.136i)27-s + (0.742 + 0.742i)29-s − 0.762·31-s + ⋯

Functional equation

Λ(s)=(768s/2 ΓC(s) L(s)=((−0.382+0.923i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(768s/2 ΓC(s+1/2) L(s)=((−0.382+0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 768768    =    28⋅32^{8} \cdot 3
Sign: −0.382+0.923i-0.382 + 0.923i
Analytic conductor: 6.132516.13251
Root analytic conductor: 2.476392.47639
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ768(577,⋅)\chi_{768} (577, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 768, ( :1/2), −0.382+0.923i)(2,\ 768,\ (\ :1/2),\ -0.382 + 0.923i)

Particular Values

L(1)L(1) ≈\approx 0.846991−1.26761i0.846991 - 1.26761i
L(12)L(\frac12) ≈\approx 0.846991−1.26761i0.846991 - 1.26761i
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+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
good5 1+(−2+2i)T−5iT2 1 + (-2 + 2i)T - 5iT^{2}
7 1+4.24iT−7T2 1 + 4.24iT - 7T^{2}
11 1+(−2.82+2.82i)T−11iT2 1 + (-2.82 + 2.82i)T - 11iT^{2}
13 1+(−3−3i)T+13iT2 1 + (-3 - 3i)T + 13iT^{2}
17 1+6T+17T2 1 + 6T + 17T^{2}
19 1+(1.41+1.41i)T+19iT2 1 + (1.41 + 1.41i)T + 19iT^{2}
23 1−2.82iT−23T2 1 - 2.82iT - 23T^{2}
29 1+(−4−4i)T+29iT2 1 + (-4 - 4i)T + 29iT^{2}
31 1+4.24T+31T2 1 + 4.24T + 31T^{2}
37 1+(−3+3i)T−37iT2 1 + (-3 + 3i)T - 37iT^{2}
41 1+10iT−41T2 1 + 10iT - 41T^{2}
43 1+(4.24−4.24i)T−43iT2 1 + (4.24 - 4.24i)T - 43iT^{2}
47 1−2.82T+47T2 1 - 2.82T + 47T^{2}
53 1+(−4+4i)T−53iT2 1 + (-4 + 4i)T - 53iT^{2}
59 1−59iT2 1 - 59iT^{2}
61 1+(−3−3i)T+61iT2 1 + (-3 - 3i)T + 61iT^{2}
67 1+(−2.82−2.82i)T+67iT2 1 + (-2.82 - 2.82i)T + 67iT^{2}
71 1+2.82iT−71T2 1 + 2.82iT - 71T^{2}
73 1−16iT−73T2 1 - 16iT - 73T^{2}
79 1−4.24T+79T2 1 - 4.24T + 79T^{2}
83 1+(11.3+11.3i)T+83iT2 1 + (11.3 + 11.3i)T + 83iT^{2}
89 1+14iT−89T2 1 + 14iT - 89T^{2}
97 1+4T+97T2 1 + 4T + 97T^{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

−10.11510962402942557345311230522, −9.017019754242029093189307405784, −8.629996951329103893467879870441, −7.17541090524028005340860416239, −6.60530406817808809405330585777, −5.72070719485549474035647467450, −4.55333170601221371658323845706, −3.77575529505352704074050354129, −1.79390694285162107155781629995, −0.851831776320703403346591185809, 1.97386060038264469952212255906, 2.86337915081147267109893599540, 4.28657100691365784008816074475, 5.41853600021421866967305718659, 6.31227264158126278770763844670, 6.59514634713179358768277490491, 8.240738928944554838331361457835, 9.056803188338565437820075487517, 9.750335414596034369890949145718, 10.56206173434048164945911644265

Graph of the ZZ-function along the critical line