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Properties

Label 2-3920-35.32-c0-0-3
Degree $2$
Conductor $3920$
Sign $-0.816 + 0.577i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 − 0.965i)3-s + (0.258 − 0.965i)5-s + (−0.5 + 0.866i)11-s + (0.707 − 0.707i)13-s − 15-s + (−0.965 + 0.258i)17-s + (1.22 − 0.707i)19-s + (−1.36 − 0.366i)23-s + (−0.866 − 0.499i)25-s + (−0.707 − 0.707i)27-s − i·29-s + (0.965 + 0.258i)33-s + (−0.366 + 1.36i)37-s + (−0.866 − 0.500i)39-s + (1 − i)43-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)3-s + (0.258 − 0.965i)5-s + (−0.5 + 0.866i)11-s + (0.707 − 0.707i)13-s − 15-s + (−0.965 + 0.258i)17-s + (1.22 − 0.707i)19-s + (−1.36 − 0.366i)23-s + (−0.866 − 0.499i)25-s + (−0.707 − 0.707i)27-s − i·29-s + (0.965 + 0.258i)33-s + (−0.366 + 1.36i)37-s + (−0.866 − 0.500i)39-s + (1 − i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-0.816 + 0.577i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ -0.816 + 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.080473708\)
\(L(\frac12)\) \(\approx\) \(1.080473708\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.258 + 0.965i)T \)
7 \( 1 \)
good3 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
17 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1 + i)T - iT^{2} \)
47 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.202492674850245152480639906290, −7.68012771458933571892261413888, −6.89224088051277476379158626322, −6.14109281290520340855595817891, −5.47997331560394578785224409249, −4.64399246901697185698409589317, −3.86249678224862480453554683134, −2.45882652140681724943220499133, −1.70189108798486642703206418787, −0.62656776523311349125675141608, 1.66612188616610596524962516862, 2.79682678977230978284815060116, 3.68228156418811431299079819086, 4.20473898764533434338325832954, 5.36321182574496830071478224450, 5.83704880813550423879337408986, 6.65239820525808608493212205836, 7.48188480289338370908835309613, 8.165644018708000655281991389193, 9.338175145840078929839517213260

Graph of the $Z$-function along the critical line