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Properties

Label 2-1920-80.69-c1-0-27
Degree $2$
Conductor $1920$
Sign $0.189 - 0.981i$
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  + (0.707 + 0.707i)3-s + (1.03 + 1.98i)5-s + 3.91·7-s + 1.00i·9-s + (2.93 + 2.93i)11-s + (−0.732 − 0.732i)13-s + (−0.674 + 2.13i)15-s + 2.89i·17-s + (−1.67 + 1.67i)19-s + (2.77 + 2.77i)21-s + 1.73·23-s + (−2.87 + 4.09i)25-s + (−0.707 + 0.707i)27-s + (4.99 − 4.99i)29-s − 10.8·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.461 + 0.887i)5-s + 1.48·7-s + 0.333i·9-s + (0.884 + 0.884i)11-s + (−0.203 − 0.203i)13-s + (−0.174 + 0.550i)15-s + 0.701i·17-s + (−0.384 + 0.384i)19-s + (0.604 + 0.604i)21-s + 0.361·23-s + (−0.574 + 0.818i)25-s + (−0.136 + 0.136i)27-s + (0.926 − 0.926i)29-s − 1.94·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.189 - 0.981i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ 0.189 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.708093173\)
\(L(\frac12)\) \(\approx\) \(2.708093173\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-1.03 - 1.98i)T \)
good7 \( 1 - 3.91T + 7T^{2} \)
11 \( 1 + (-2.93 - 2.93i)T + 11iT^{2} \)
13 \( 1 + (0.732 + 0.732i)T + 13iT^{2} \)
17 \( 1 - 2.89iT - 17T^{2} \)
19 \( 1 + (1.67 - 1.67i)T - 19iT^{2} \)
23 \( 1 - 1.73T + 23T^{2} \)
29 \( 1 + (-4.99 + 4.99i)T - 29iT^{2} \)
31 \( 1 + 10.8T + 31T^{2} \)
37 \( 1 + (-6.41 + 6.41i)T - 37iT^{2} \)
41 \( 1 + 0.00577iT - 41T^{2} \)
43 \( 1 + (2.23 - 2.23i)T - 43iT^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 + (-5.55 + 5.55i)T - 53iT^{2} \)
59 \( 1 + (3.83 + 3.83i)T + 59iT^{2} \)
61 \( 1 + (9.30 - 9.30i)T - 61iT^{2} \)
67 \( 1 + (3.85 + 3.85i)T + 67iT^{2} \)
71 \( 1 + 1.15iT - 71T^{2} \)
73 \( 1 - 7.98T + 73T^{2} \)
79 \( 1 - 0.843T + 79T^{2} \)
83 \( 1 + (5.20 + 5.20i)T + 83iT^{2} \)
89 \( 1 - 5.40iT - 89T^{2} \)
97 \( 1 + 2.24iT - 97T^{2} \)
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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

−9.396382068225693683118656881321, −8.626375311382151831087986332634, −7.77401506438046097776667280934, −7.17481729447702965201313724515, −6.19046763614398748715390521816, −5.28837336404672834707358927758, −4.36718826170293146428577924539, −3.64685200114503308318387753557, −2.28629099722767270748724421301, −1.68355909812362057246284718905, 1.02795096971622481846785532630, 1.73989019949288319503705658720, 2.93897251521751222627460148562, 4.26249449544980511044899258184, 4.91448259801028392637053227897, 5.76266811119195275910815761947, 6.70456464057873781567744477658, 7.63851852076306172274711468022, 8.342665999098747695896880724095, 8.998233140250186650957637827026

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