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Properties

Label 2-1920-80.43-c1-0-32
Degree $2$
Conductor $1920$
Sign $-0.0208 + 0.999i$
Analytic cond. $15.3312$
Root an. cond. $3.91551$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (−1.54 + 1.61i)5-s + (−0.143 + 0.143i)7-s − 9-s + (0.749 − 0.749i)11-s − 3.29·13-s + (−1.61 − 1.54i)15-s + (1.35 − 1.35i)17-s + (−4.25 + 4.25i)19-s + (−0.143 − 0.143i)21-s + (−0.837 − 0.837i)23-s + (−0.207 − 4.99i)25-s − i·27-s + (−2.77 − 2.77i)29-s − 6.60i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.692 + 0.721i)5-s + (−0.0543 + 0.0543i)7-s − 0.333·9-s + (0.225 − 0.225i)11-s − 0.912·13-s + (−0.416 − 0.399i)15-s + (0.329 − 0.329i)17-s + (−0.976 + 0.976i)19-s + (−0.0314 − 0.0314i)21-s + (−0.174 − 0.174i)23-s + (−0.0414 − 0.999i)25-s − 0.192i·27-s + (−0.515 − 0.515i)29-s − 1.18i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0208 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0208 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.0208 + 0.999i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (1183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ -0.0208 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3789577767\)
\(L(\frac12)\) \(\approx\) \(0.3789577767\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
5 \( 1 + (1.54 - 1.61i)T \)
good7 \( 1 + (0.143 - 0.143i)T - 7iT^{2} \)
11 \( 1 + (-0.749 + 0.749i)T - 11iT^{2} \)
13 \( 1 + 3.29T + 13T^{2} \)
17 \( 1 + (-1.35 + 1.35i)T - 17iT^{2} \)
19 \( 1 + (4.25 - 4.25i)T - 19iT^{2} \)
23 \( 1 + (0.837 + 0.837i)T + 23iT^{2} \)
29 \( 1 + (2.77 + 2.77i)T + 29iT^{2} \)
31 \( 1 + 6.60iT - 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 - 1.72iT - 41T^{2} \)
43 \( 1 + 4.17T + 43T^{2} \)
47 \( 1 + (8.54 + 8.54i)T + 47iT^{2} \)
53 \( 1 - 5.05iT - 53T^{2} \)
59 \( 1 + (3.08 + 3.08i)T + 59iT^{2} \)
61 \( 1 + (-5.00 + 5.00i)T - 61iT^{2} \)
67 \( 1 + 4.26T + 67T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + (-11.6 + 11.6i)T - 73iT^{2} \)
79 \( 1 - 9.95T + 79T^{2} \)
83 \( 1 + 10.0iT - 83T^{2} \)
89 \( 1 + 5.76T + 89T^{2} \)
97 \( 1 + (-11.7 + 11.7i)T - 97iT^{2} \)
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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

−9.119229868568539879310408292662, −7.993981470019477159256076671309, −7.69713788123999066337629758271, −6.52832677030834343580648084774, −5.92301849556323191970251910349, −4.73026843897808724532759811059, −4.02753170429768365490857446118, −3.17009779638607081868382886223, −2.18169484218857003243017764520, −0.14455063903767439666735515740, 1.22104948820808250780662726806, 2.42960471773013396980181043694, 3.59687783302242672243542214989, 4.59510104696982343268878000882, 5.24086787188664586601698599744, 6.39297224694644375472934318153, 7.12532283312195097939543937362, 7.83233817770745257680117012047, 8.546870815839127591280652742369, 9.240548005719134966667134989708

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