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Properties

Label 2-40e2-20.3-c1-0-17
Degree 22
Conductor 16001600
Sign 0.850−0.525i0.850 - 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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 2i)3-s + (2 + 2i)7-s − 5i·9-s + (−1 − i)13-s + (5 − 5i)17-s + 4·19-s − 8·21-s + (2 − 2i)23-s + (4 + 4i)27-s − 4i·29-s − 4i·31-s + (1 − i)37-s + 4·39-s + (6 − 6i)43-s + (−2 − 2i)47-s + ⋯
L(s)  = 1  + (−1.15 + 1.15i)3-s + (0.755 + 0.755i)7-s − 1.66i·9-s + (−0.277 − 0.277i)13-s + (1.21 − 1.21i)17-s + 0.917·19-s − 1.74·21-s + (0.417 − 0.417i)23-s + (0.769 + 0.769i)27-s − 0.742i·29-s − 0.718i·31-s + (0.164 − 0.164i)37-s + 0.640·39-s + (0.914 − 0.914i)43-s + (−0.291 − 0.291i)47-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: 0.850−0.525i0.850 - 0.525i
Analytic conductor: 12.776012.7760
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1600(1343,⋅)\chi_{1600} (1343, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1600, ( :1/2), 0.850−0.525i)(2,\ 1600,\ (\ :1/2),\ 0.850 - 0.525i)

Particular Values

L(1)L(1) ≈\approx 1.2369088781.236908878
L(12)L(\frac12) ≈\approx 1.2369088781.236908878
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+(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

−9.824035746050276995318001059359, −8.879497359957271322294128083006, −7.910183342301211870629676942020, −7.00621352936811647541120539422, −5.76375537033056747436714548833, −5.37128852542097409721740056601, −4.75438873219656222026554312231, −3.71702910636198709638027100023, −2.54172334288598827486348885621, −0.74224336133570552787174303818, 1.04196385229980683940419055957, 1.64414702522611555776394123864, 3.29613539383094573066654059563, 4.58476895883111918494663699182, 5.37475069642031071734892128175, 6.10907772588770603036871078024, 6.99354812874300665672723799699, 7.61727826073966844802561949689, 8.133125465690917920606161970208, 9.413622223939550624798789734255

Graph of the ZZ-function along the critical line