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Properties

Label 2-4032-48.11-c1-0-12
Degree 22
Conductor 40324032
Sign −0.417−0.908i-0.417 - 0.908i
Analytic cond. 32.195632.1956
Root an. cond. 5.674125.67412
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  + (−1.65 + 1.65i)5-s − 7-s + (0.993 + 0.993i)11-s + (2.62 − 2.62i)13-s + 2.77i·17-s + (−1.56 − 1.56i)19-s − 1.05i·23-s − 0.491i·25-s + (3.47 + 3.47i)29-s − 1.06i·31-s + (1.65 − 1.65i)35-s + (−0.0657 − 0.0657i)37-s + 6.31·41-s + (2.38 − 2.38i)43-s + 1.47·47-s + ⋯
L(s)  = 1  + (−0.741 + 0.741i)5-s − 0.377·7-s + (0.299 + 0.299i)11-s + (0.728 − 0.728i)13-s + 0.673i·17-s + (−0.358 − 0.358i)19-s − 0.219i·23-s − 0.0982i·25-s + (0.645 + 0.645i)29-s − 0.190i·31-s + (0.280 − 0.280i)35-s + (−0.0108 − 0.0108i)37-s + 0.985·41-s + (0.364 − 0.364i)43-s + 0.214·47-s + ⋯

Functional equation

Λ(s)=(4032s/2 ΓC(s) L(s)=((−0.417−0.908i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4032s/2 ΓC(s+1/2) L(s)=((−0.417−0.908i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40324032    =    26⋅32⋅72^{6} \cdot 3^{2} \cdot 7
Sign: −0.417−0.908i-0.417 - 0.908i
Analytic conductor: 32.195632.1956
Root analytic conductor: 5.674125.67412
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4032(3599,⋅)\chi_{4032} (3599, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4032, ( :1/2), −0.417−0.908i)(2,\ 4032,\ (\ :1/2),\ -0.417 - 0.908i)

Particular Values

L(1)L(1) ≈\approx 1.0976973521.097697352
L(12)L(\frac12) ≈\approx 1.0976973521.097697352
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
3 1 1
7 1+T 1 + T
good5 1+(1.65−1.65i)T−5iT2 1 + (1.65 - 1.65i)T - 5iT^{2}
11 1+(−0.993−0.993i)T+11iT2 1 + (-0.993 - 0.993i)T + 11iT^{2}
13 1+(−2.62+2.62i)T−13iT2 1 + (-2.62 + 2.62i)T - 13iT^{2}
17 1−2.77iT−17T2 1 - 2.77iT - 17T^{2}
19 1+(1.56+1.56i)T+19iT2 1 + (1.56 + 1.56i)T + 19iT^{2}
23 1+1.05iT−23T2 1 + 1.05iT - 23T^{2}
29 1+(−3.47−3.47i)T+29iT2 1 + (-3.47 - 3.47i)T + 29iT^{2}
31 1+1.06iT−31T2 1 + 1.06iT - 31T^{2}
37 1+(0.0657+0.0657i)T+37iT2 1 + (0.0657 + 0.0657i)T + 37iT^{2}
41 1−6.31T+41T2 1 - 6.31T + 41T^{2}
43 1+(−2.38+2.38i)T−43iT2 1 + (-2.38 + 2.38i)T - 43iT^{2}
47 1−1.47T+47T2 1 - 1.47T + 47T^{2}
53 1+(7.63−7.63i)T−53iT2 1 + (7.63 - 7.63i)T - 53iT^{2}
59 1+(−4.15−4.15i)T+59iT2 1 + (-4.15 - 4.15i)T + 59iT^{2}
61 1+(7.78−7.78i)T−61iT2 1 + (7.78 - 7.78i)T - 61iT^{2}
67 1+(1.98+1.98i)T+67iT2 1 + (1.98 + 1.98i)T + 67iT^{2}
71 1+13.0iT−71T2 1 + 13.0iT - 71T^{2}
73 1−9.50iT−73T2 1 - 9.50iT - 73T^{2}
79 1+9.85iT−79T2 1 + 9.85iT - 79T^{2}
83 1+(1.13−1.13i)T−83iT2 1 + (1.13 - 1.13i)T - 83iT^{2}
89 1−7.04T+89T2 1 - 7.04T + 89T^{2}
97 1−5.35T+97T2 1 - 5.35T + 97T^{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

−8.703677991825013504911564430843, −7.78236322160435729889830105464, −7.33038063638632771137248481569, −6.40931165747377994361529072165, −5.97296369873004281842333645455, −4.81760478539520796601869093086, −3.94863763989858136991493618883, −3.32329317624028099415861503859, −2.48334420805933749804018656527, −1.10160626949482344769297383339, 0.36835501851288102039301171742, 1.45806762928041246259225830986, 2.73118433618980860955347271339, 3.76151758725359021621303743991, 4.29660689256743173541367549963, 5.11265075424814277895089051361, 6.12231307248142144448581387870, 6.64669305287690372370869646405, 7.61855224847689593719114793847, 8.251525131939462300682962216450

Graph of the ZZ-function along the critical line