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Properties

Label 2-1920-1920.1589-c0-0-1
Degree $2$
Conductor $1920$
Sign $-0.941 + 0.336i$
Analytic cond. $0.958204$
Root an. cond. $0.978879$
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  + (0.290 − 0.956i)2-s + (0.0980 + 0.995i)3-s + (−0.831 − 0.555i)4-s + (−0.881 − 0.471i)5-s + (0.980 + 0.195i)6-s + (−0.773 + 0.634i)8-s + (−0.980 + 0.195i)9-s + (−0.707 + 0.707i)10-s + (0.471 − 0.881i)12-s + (0.382 − 0.923i)15-s + (0.382 + 0.923i)16-s + (−0.360 − 0.871i)17-s + (−0.0980 + 0.995i)18-s + (0.448 − 1.47i)19-s + (0.471 + 0.881i)20-s + ⋯
L(s)  = 1  + (0.290 − 0.956i)2-s + (0.0980 + 0.995i)3-s + (−0.831 − 0.555i)4-s + (−0.881 − 0.471i)5-s + (0.980 + 0.195i)6-s + (−0.773 + 0.634i)8-s + (−0.980 + 0.195i)9-s + (−0.707 + 0.707i)10-s + (0.471 − 0.881i)12-s + (0.382 − 0.923i)15-s + (0.382 + 0.923i)16-s + (−0.360 − 0.871i)17-s + (−0.0980 + 0.995i)18-s + (0.448 − 1.47i)19-s + (0.471 + 0.881i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.941 + 0.336i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (1589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :0),\ -0.941 + 0.336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5276777826\)
\(L(\frac12)\) \(\approx\) \(0.5276777826\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.290 + 0.956i)T \)
3 \( 1 + (-0.0980 - 0.995i)T \)
5 \( 1 + (0.881 + 0.471i)T \)
good7 \( 1 + (0.923 + 0.382i)T^{2} \)
11 \( 1 + (0.195 - 0.980i)T^{2} \)
13 \( 1 + (0.555 - 0.831i)T^{2} \)
17 \( 1 + (0.360 + 0.871i)T + (-0.707 + 0.707i)T^{2} \)
19 \( 1 + (-0.448 + 1.47i)T + (-0.831 - 0.555i)T^{2} \)
23 \( 1 + (1.59 + 1.06i)T + (0.382 + 0.923i)T^{2} \)
29 \( 1 + (-0.195 - 0.980i)T^{2} \)
31 \( 1 + (1.17 + 1.17i)T + iT^{2} \)
37 \( 1 + (-0.831 + 0.555i)T^{2} \)
41 \( 1 + (0.382 + 0.923i)T^{2} \)
43 \( 1 + (0.980 + 0.195i)T^{2} \)
47 \( 1 + (1.83 - 0.761i)T + (0.707 - 0.707i)T^{2} \)
53 \( 1 + (0.301 - 0.247i)T + (0.195 - 0.980i)T^{2} \)
59 \( 1 + (0.555 + 0.831i)T^{2} \)
61 \( 1 + (-1.26 + 0.124i)T + (0.980 - 0.195i)T^{2} \)
67 \( 1 + (-0.980 + 0.195i)T^{2} \)
71 \( 1 + (-0.923 - 0.382i)T^{2} \)
73 \( 1 + (0.923 - 0.382i)T^{2} \)
79 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
83 \( 1 + (-1.87 - 0.569i)T + (0.831 + 0.555i)T^{2} \)
89 \( 1 + (-0.382 + 0.923i)T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.341233890264965569172119638188, −8.502577613178969859661558014205, −7.83272653017422601514878818753, −6.50008310946497470168400019603, −5.32836032595504352129068504972, −4.72296883819786786179717371795, −4.08279286687123226701748298128, −3.24488204389432868335952814816, −2.28243356782316659560187739051, −0.33752620505131144918693475944, 1.77649817962920882464801964288, 3.39825244546996647850299339220, 3.77321542292589466351670339441, 5.13658745463758406661343743544, 6.02393786499147082075937218000, 6.61842445307700238822214875808, 7.44397252364453166230592184291, 8.029581844724515501228698718632, 8.410480722255884649605470771365, 9.503028255640614495664100111077

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