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Properties

Label 2-640-40.3-c1-0-6
Degree 22
Conductor 640640
Sign 0.229−0.973i0.229 - 0.973i
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 + 2i)3-s + (−2 − i)5-s + (2 + 2i)7-s + 5i·9-s + 4·11-s + (3 − 3i)13-s + (−2 − 6i)15-s + (−3 + 3i)17-s + 8i·21-s + (−6 + 6i)23-s + (3 + 4i)25-s + (−4 + 4i)27-s − 2·29-s − 4i·31-s + (8 + 8i)33-s + ⋯
L(s)  = 1  + (1.15 + 1.15i)3-s + (−0.894 − 0.447i)5-s + (0.755 + 0.755i)7-s + 1.66i·9-s + 1.20·11-s + (0.832 − 0.832i)13-s + (−0.516 − 1.54i)15-s + (−0.727 + 0.727i)17-s + 1.74i·21-s + (−1.25 + 1.25i)23-s + (0.600 + 0.800i)25-s + (−0.769 + 0.769i)27-s − 0.371·29-s − 0.718i·31-s + (1.39 + 1.39i)33-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 640640    =    27⋅52^{7} \cdot 5
Sign: 0.229−0.973i0.229 - 0.973i
Analytic conductor: 5.110425.11042
Root analytic conductor: 2.260622.26062
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ640(63,⋅)\chi_{640} (63, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 640, ( :1/2), 0.229−0.973i)(2,\ 640,\ (\ :1/2),\ 0.229 - 0.973i)

Particular Values

L(1)L(1) ≈\approx 1.64644+1.30302i1.64644 + 1.30302i
L(12)L(\frac12) ≈\approx 1.64644+1.30302i1.64644 + 1.30302i
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+(−2−2i)T+3iT2 1 + (-2 - 2i)T + 3iT^{2}
7 1+(−2−2i)T+7iT2 1 + (-2 - 2i)T + 7iT^{2}
11 1−4T+11T2 1 - 4T + 11T^{2}
13 1+(−3+3i)T−13iT2 1 + (-3 + 3i)T - 13iT^{2}
17 1+(3−3i)T−17iT2 1 + (3 - 3i)T - 17iT^{2}
19 1−19T2 1 - 19T^{2}
23 1+(6−6i)T−23iT2 1 + (6 - 6i)T - 23iT^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1+4iT−31T2 1 + 4iT - 31T^{2}
37 1+(−3−3i)T+37iT2 1 + (-3 - 3i)T + 37iT^{2}
41 1+41T2 1 + 41T^{2}
43 1+(6+6i)T+43iT2 1 + (6 + 6i)T + 43iT^{2}
47 1+(−6−6i)T+47iT2 1 + (-6 - 6i)T + 47iT^{2}
53 1+(3−3i)T−53iT2 1 + (3 - 3i)T - 53iT^{2}
59 1+8iT−59T2 1 + 8iT - 59T^{2}
61 1+6iT−61T2 1 + 6iT - 61T^{2}
67 1+(−6+6i)T−67iT2 1 + (-6 + 6i)T - 67iT^{2}
71 1+12iT−71T2 1 + 12iT - 71T^{2}
73 1+(5+5i)T+73iT2 1 + (5 + 5i)T + 73iT^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1+(−6−6i)T+83iT2 1 + (-6 - 6i)T + 83iT^{2}
89 1−89T2 1 - 89T^{2}
97 1+(−11+11i)T−97iT2 1 + (-11 + 11i)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

−10.78000155244330814722909761744, −9.595837863586471341676557871521, −8.973402191308713849250181184958, −8.277194263967601432643402914382, −7.80462613316474300730407221303, −6.11135016692257192349242371734, −4.93993805467126066029941428681, −3.97752863634768898288480899700, −3.46382502047251490782923274322, −1.85686263916154129754932742731, 1.16603695945558309403988358106, 2.37312902068764667396529395229, 3.74578683485653843943465030451, 4.34712349221846685579644102941, 6.47946745621833698467095916299, 6.93457328775907997463927279719, 7.76502993630664451722651564132, 8.507285587563172611323430280318, 9.106066377149385055643637010456, 10.49953090973320678114938046979

Graph of the ZZ-function along the critical line