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Properties

Label 2-1920-80.29-c1-0-40
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: $\chi_{1920} (289, \cdot )$
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

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

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