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Properties

Label 2-640-40.27-c1-0-12
Degree 22
Conductor 640640
Sign 0.973−0.229i0.973 - 0.229i
Analytic cond. 5.110425.11042
Root an. cond. 2.260622.26062
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  + (2 − i)5-s + 3i·9-s + (5 + 5i)13-s + (−3 − 3i)17-s + (3 − 4i)25-s + 10·29-s + (5 − 5i)37-s − 8·41-s + (3 + 6i)45-s + 7i·49-s + (−5 − 5i)53-s + 10i·61-s + (15 + 5i)65-s + (11 − 11i)73-s − 9·81-s + ⋯
L(s)  = 1  + (0.894 − 0.447i)5-s + i·9-s + (1.38 + 1.38i)13-s + (−0.727 − 0.727i)17-s + (0.600 − 0.800i)25-s + 1.85·29-s + (0.821 − 0.821i)37-s − 1.24·41-s + (0.447 + 0.894i)45-s + i·49-s + (−0.686 − 0.686i)53-s + 1.28i·61-s + (1.86 + 0.620i)65-s + (1.28 − 1.28i)73-s − 81-s + ⋯

Functional equation

Λ(s)=(640s/2 ΓC(s) L(s)=((0.973−0.229i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(640s/2 ΓC(s+1/2) L(s)=((0.973−0.229i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 640640    =    27⋅52^{7} \cdot 5
Sign: 0.973−0.229i0.973 - 0.229i
Analytic conductor: 5.110425.11042
Root analytic conductor: 2.260622.26062
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ640(447,⋅)\chi_{640} (447, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 640, ( :1/2), 0.973−0.229i)(2,\ 640,\ (\ :1/2),\ 0.973 - 0.229i)

Particular Values

L(1)L(1) ≈\approx 1.79097+0.208530i1.79097 + 0.208530i
L(12)L(\frac12) ≈\approx 1.79097+0.208530i1.79097 + 0.208530i
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
5 1+(−2+i)T 1 + (-2 + i)T
good3 1−3iT2 1 - 3iT^{2}
7 1−7iT2 1 - 7iT^{2}
11 1+11T2 1 + 11T^{2}
13 1+(−5−5i)T+13iT2 1 + (-5 - 5i)T + 13iT^{2}
17 1+(3+3i)T+17iT2 1 + (3 + 3i)T + 17iT^{2}
19 1−19T2 1 - 19T^{2}
23 1+23iT2 1 + 23iT^{2}
29 1−10T+29T2 1 - 10T + 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1+(−5+5i)T−37iT2 1 + (-5 + 5i)T - 37iT^{2}
41 1+8T+41T2 1 + 8T + 41T^{2}
43 1−43iT2 1 - 43iT^{2}
47 1−47iT2 1 - 47iT^{2}
53 1+(5+5i)T+53iT2 1 + (5 + 5i)T + 53iT^{2}
59 1−59T2 1 - 59T^{2}
61 1−10iT−61T2 1 - 10iT - 61T^{2}
67 1+67iT2 1 + 67iT^{2}
71 1−71T2 1 - 71T^{2}
73 1+(−11+11i)T−73iT2 1 + (-11 + 11i)T - 73iT^{2}
79 1+79T2 1 + 79T^{2}
83 1−83iT2 1 - 83iT^{2}
89 1−16iT−89T2 1 - 16iT - 89T^{2}
97 1+(13+13i)T+97iT2 1 + (13 + 13i)T + 97iT^{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

−10.65767338927181772321516981806, −9.656067500408260093938161875664, −8.884489111685687606360383364266, −8.208456564630146961974742584140, −6.86542742522800213775067869616, −6.16604991613569018299740908008, −5.04073816145963120921586494585, −4.25650204458828518379193223841, −2.58447357404155016141704038582, −1.47774598846936886897286320391, 1.20775996873364005314871711799, 2.81448259458608918789009487815, 3.74490299329917498493524187894, 5.19857404185808768269645006621, 6.31517888356130832358685866429, 6.54182879850666127986144728761, 8.116407126842502297632691688064, 8.766128671741601924661907876758, 9.816763381046881515957460382662, 10.47327371233460928634585879200

Graph of the ZZ-function along the critical line