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Properties

Label 2-40e2-20.7-c1-0-28
Degree 22
Conductor 16001600
Sign −0.850−0.525i-0.850 - 0.525i
Analytic cond. 12.776012.7760
Root an. cond. 3.574363.57436
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)3-s + (−3 + 3i)7-s − i·9-s − 2i·11-s + (3 − 3i)13-s + (−1 − i)17-s + 4·19-s + 6·21-s + (−1 − i)23-s + (−4 + 4i)27-s + 10i·31-s + (−2 + 2i)33-s + (−1 − i)37-s − 6·39-s − 10·41-s + ⋯
L(s)  = 1  + (−0.577 − 0.577i)3-s + (−1.13 + 1.13i)7-s − 0.333i·9-s − 0.603i·11-s + (0.832 − 0.832i)13-s + (−0.242 − 0.242i)17-s + 0.917·19-s + 1.30·21-s + (−0.208 − 0.208i)23-s + (−0.769 + 0.769i)27-s + 1.79i·31-s + (−0.348 + 0.348i)33-s + (−0.164 − 0.164i)37-s − 0.960·39-s − 1.56·41-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=((−0.850−0.525i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+1/2) L(s)=((−0.850−0.525i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: −0.850−0.525i-0.850 - 0.525i
Analytic conductor: 12.776012.7760
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1600(1407,⋅)\chi_{1600} (1407, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 1600, ( :1/2), −0.850−0.525i)(2,\ 1600,\ (\ :1/2),\ -0.850 - 0.525i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1
good3 1+(1+i)T+3iT2 1 + (1 + i)T + 3iT^{2}
7 1+(3−3i)T−7iT2 1 + (3 - 3i)T - 7iT^{2}
11 1+2iT−11T2 1 + 2iT - 11T^{2}
13 1+(−3+3i)T−13iT2 1 + (-3 + 3i)T - 13iT^{2}
17 1+(1+i)T+17iT2 1 + (1 + i)T + 17iT^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1+(1+i)T+23iT2 1 + (1 + i)T + 23iT^{2}
29 1−29T2 1 - 29T^{2}
31 1−10iT−31T2 1 - 10iT - 31T^{2}
37 1+(1+i)T+37iT2 1 + (1 + i)T + 37iT^{2}
41 1+10T+41T2 1 + 10T + 41T^{2}
43 1+(5+5i)T+43iT2 1 + (5 + 5i)T + 43iT^{2}
47 1+(3−3i)T−47iT2 1 + (3 - 3i)T - 47iT^{2}
53 1+(5−5i)T−53iT2 1 + (5 - 5i)T - 53iT^{2}
59 1+12T+59T2 1 + 12T + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1+(−1+i)T−67iT2 1 + (-1 + i)T - 67iT^{2}
71 1−2iT−71T2 1 - 2iT - 71T^{2}
73 1+(1−i)T−73iT2 1 + (1 - i)T - 73iT^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1+(5+5i)T+83iT2 1 + (5 + 5i)T + 83iT^{2}
89 1−16iT−89T2 1 - 16iT - 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

−8.912927045451480904730248988512, −8.283539223919954088699060381263, −7.08197431740022071042206041486, −6.40693536831449581400013557028, −5.81728526750977657581662811718, −5.14296477111565930907404971565, −3.43677035799624185483612676385, −3.00867318863603356950809490039, −1.38134933259869946030630000976, 0, 1.68082192265665586254738887024, 3.26524033774252583616087916396, 4.08010266609314751459190029639, 4.76596156559381746265679850396, 5.89290166053203571505447211701, 6.60444173453214833013871216286, 7.35466130558282302581545567147, 8.238087443664100648297461376930, 9.492720387548640946458888795814

Graph of the ZZ-function along the critical line