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Properties

Label 2-1920-1920.1229-c0-0-1
Degree $2$
Conductor $1920$
Sign $0.427 + 0.903i$
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  + (−0.634 − 0.773i)2-s + (0.956 − 0.290i)3-s + (−0.195 + 0.980i)4-s + (−0.995 + 0.0980i)5-s + (−0.831 − 0.555i)6-s + (0.881 − 0.471i)8-s + (0.831 − 0.555i)9-s + (0.707 + 0.707i)10-s + (0.0980 + 0.995i)12-s + (−0.923 + 0.382i)15-s + (−0.923 − 0.382i)16-s + (0.181 + 0.0750i)17-s + (−0.956 − 0.290i)18-s + (1.11 + 1.36i)19-s + (0.0980 − 0.995i)20-s + ⋯
L(s)  = 1  + (−0.634 − 0.773i)2-s + (0.956 − 0.290i)3-s + (−0.195 + 0.980i)4-s + (−0.995 + 0.0980i)5-s + (−0.831 − 0.555i)6-s + (0.881 − 0.471i)8-s + (0.831 − 0.555i)9-s + (0.707 + 0.707i)10-s + (0.0980 + 0.995i)12-s + (−0.923 + 0.382i)15-s + (−0.923 − 0.382i)16-s + (0.181 + 0.0750i)17-s + (−0.956 − 0.290i)18-s + (1.11 + 1.36i)19-s + (0.0980 − 0.995i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.427 + 0.903i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (1229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :0),\ 0.427 + 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.018600579\)
\(L(\frac12)\) \(\approx\) \(1.018600579\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.634 + 0.773i)T \)
3 \( 1 + (-0.956 + 0.290i)T \)
5 \( 1 + (0.995 - 0.0980i)T \)
good7 \( 1 + (0.382 + 0.923i)T^{2} \)
11 \( 1 + (0.555 - 0.831i)T^{2} \)
13 \( 1 + (0.980 + 0.195i)T^{2} \)
17 \( 1 + (-0.181 - 0.0750i)T + (0.707 + 0.707i)T^{2} \)
19 \( 1 + (-1.11 - 1.36i)T + (-0.195 + 0.980i)T^{2} \)
23 \( 1 + (-0.301 + 1.51i)T + (-0.923 - 0.382i)T^{2} \)
29 \( 1 + (-0.555 - 0.831i)T^{2} \)
31 \( 1 + (-0.275 + 0.275i)T - iT^{2} \)
37 \( 1 + (-0.195 - 0.980i)T^{2} \)
41 \( 1 + (-0.923 - 0.382i)T^{2} \)
43 \( 1 + (-0.831 - 0.555i)T^{2} \)
47 \( 1 + (0.222 - 0.536i)T + (-0.707 - 0.707i)T^{2} \)
53 \( 1 + (-0.979 + 0.523i)T + (0.555 - 0.831i)T^{2} \)
59 \( 1 + (0.980 - 0.195i)T^{2} \)
61 \( 1 + (-0.273 - 0.902i)T + (-0.831 + 0.555i)T^{2} \)
67 \( 1 + (0.831 - 0.555i)T^{2} \)
71 \( 1 + (-0.382 - 0.923i)T^{2} \)
73 \( 1 + (0.382 - 0.923i)T^{2} \)
79 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
83 \( 1 + (-1.28 + 1.05i)T + (0.195 - 0.980i)T^{2} \)
89 \( 1 + (0.923 - 0.382i)T^{2} \)
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

−9.104722891514288549729153344851, −8.503275602202487649729621730952, −7.84678585022373105762068199031, −7.36641077532145490808400080968, −6.43082357025525085673245609806, −4.80141593045719154915762081663, −3.86847984486735762056836613492, −3.28669763754531592469725556305, −2.34250871460363798647719525194, −1.05985424238328155739961515779, 1.23277635877353839423090253476, 2.77393738830686480813819873954, 3.75538015291649026585630210803, 4.74949769977221128782557489708, 5.40788020389105052250072097987, 6.85317240304341620721806424356, 7.34758826983682446912340856222, 7.968114601073156765206305290463, 8.688430277386475668554513416842, 9.359177006557406671355229361133

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