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Properties

Label 2-320-20.7-c1-0-2
Degree 22
Conductor 320320
Sign 0.525−0.850i0.525 - 0.850i
Analytic cond. 2.555212.55521
Root an. cond. 1.598501.59850
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 + (1 − i)13-s + (6 + 2i)15-s + (−5 − 5i)17-s + 4·19-s − 8·21-s + (−2 − 2i)23-s + (3 − 4i)25-s + (−4 + 4i)27-s + 4i·29-s + 4i·31-s + (−2 + 6i)35-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 + (0.277 − 0.277i)13-s + (1.54 + 0.516i)15-s + (−1.21 − 1.21i)17-s + 0.917·19-s − 1.74·21-s + (−0.417 − 0.417i)23-s + (0.600 − 0.800i)25-s + (−0.769 + 0.769i)27-s + 0.742i·29-s + 0.718i·31-s + (−0.338 + 1.01i)35-s + ⋯

Functional equation

Λ(s)=(320s/2 ΓC(s) L(s)=((0.525−0.850i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(320s/2 ΓC(s+1/2) L(s)=((0.525−0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320320    =    26⋅52^{6} \cdot 5
Sign: 0.525−0.850i0.525 - 0.850i
Analytic conductor: 2.555212.55521
Root analytic conductor: 1.598501.59850
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ320(127,⋅)\chi_{320} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 320, ( :1/2), 0.525−0.850i)(2,\ 320,\ (\ :1/2),\ 0.525 - 0.850i)

Particular Values

L(1)L(1) ≈\approx 1.70817+0.952368i1.70817 + 0.952368i
L(12)L(\frac12) ≈\approx 1.70817+0.952368i1.70817 + 0.952368i
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 1−11T2 1 - 11T^{2}
13 1+(−1+i)T−13iT2 1 + (-1 + i)T - 13iT^{2}
17 1+(5+5i)T+17iT2 1 + (5 + 5i)T + 17iT^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1+(2+2i)T+23iT2 1 + (2 + 2i)T + 23iT^{2}
29 1−4iT−29T2 1 - 4iT - 29T^{2}
31 1−4iT−31T2 1 - 4iT - 31T^{2}
37 1+(1+i)T+37iT2 1 + (1 + i)T + 37iT^{2}
41 1+41T2 1 + 41T^{2}
43 1+(6+6i)T+43iT2 1 + (6 + 6i)T + 43iT^{2}
47 1+(−2+2i)T−47iT2 1 + (-2 + 2i)T - 47iT^{2}
53 1+(−7+7i)T−53iT2 1 + (-7 + 7i)T - 53iT^{2}
59 1−4T+59T2 1 - 4T + 59T^{2}
61 1−4T+61T2 1 - 4T + 61T^{2}
67 1+(10−10i)T−67iT2 1 + (10 - 10i)T - 67iT^{2}
71 1+12iT−71T2 1 + 12iT - 71T^{2}
73 1+(3−3i)T−73iT2 1 + (3 - 3i)T - 73iT^{2}
79 1+16T+79T2 1 + 16T + 79T^{2}
83 1+(2+2i)T+83iT2 1 + (2 + 2i)T + 83iT^{2}
89 1−89T2 1 - 89T^{2}
97 1+(3+3i)T+97iT2 1 + (3 + 3i)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

−11.76879059408955910240786623277, −10.40154355201536333201077454439, −9.779715327005286913427318400166, −8.933495585803860623297955552005, −8.648911364248645495779822676460, −6.97603214422178413439368945778, −5.61286764504178668945953674496, −4.69476694812651960367409068810, −3.28845776483147009444881669870, −2.36412307366741810259947605973, 1.59133414528139497970127095113, 2.75590623724216032251336061566, 3.91646776746271486704453338781, 6.02200135929139039453656588515, 6.74232386529797678794549029764, 7.55229769301900557870915055929, 8.623402503323118758598025292129, 9.524014131081033781687273619419, 10.34640463301166789617916303982, 11.57315681538910339168721901959

Graph of the ZZ-function along the critical line