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Properties

Label 2-1920-1920.509-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  + (−0.881 − 0.471i)2-s + (−0.634 − 0.773i)3-s + (0.555 + 0.831i)4-s + (0.956 + 0.290i)5-s + (0.195 + 0.980i)6-s + (−0.0980 − 0.995i)8-s + (−0.195 + 0.980i)9-s + (−0.707 − 0.707i)10-s + (0.290 − 0.956i)12-s + (−0.382 − 0.923i)15-s + (−0.382 + 0.923i)16-s + (0.222 − 0.536i)17-s + (0.634 − 0.773i)18-s + (−0.172 − 0.0924i)19-s + (0.290 + 0.956i)20-s + ⋯
L(s)  = 1  + (−0.881 − 0.471i)2-s + (−0.634 − 0.773i)3-s + (0.555 + 0.831i)4-s + (0.956 + 0.290i)5-s + (0.195 + 0.980i)6-s + (−0.0980 − 0.995i)8-s + (−0.195 + 0.980i)9-s + (−0.707 − 0.707i)10-s + (0.290 − 0.956i)12-s + (−0.382 − 0.923i)15-s + (−0.382 + 0.923i)16-s + (0.222 − 0.536i)17-s + (0.634 − 0.773i)18-s + (−0.172 − 0.0924i)19-s + (0.290 + 0.956i)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: $\chi_{1920} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :0),\ 0.427 + 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7353193127\)
\(L(\frac12)\) \(\approx\) \(0.7353193127\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.881 + 0.471i)T \)
3 \( 1 + (0.634 + 0.773i)T \)
5 \( 1 + (-0.956 - 0.290i)T \)
good7 \( 1 + (-0.923 + 0.382i)T^{2} \)
11 \( 1 + (-0.980 + 0.195i)T^{2} \)
13 \( 1 + (0.831 - 0.555i)T^{2} \)
17 \( 1 + (-0.222 + 0.536i)T + (-0.707 - 0.707i)T^{2} \)
19 \( 1 + (0.172 + 0.0924i)T + (0.555 + 0.831i)T^{2} \)
23 \( 1 + (0.523 + 0.783i)T + (-0.382 + 0.923i)T^{2} \)
29 \( 1 + (0.980 + 0.195i)T^{2} \)
31 \( 1 + (-0.785 + 0.785i)T - iT^{2} \)
37 \( 1 + (0.555 - 0.831i)T^{2} \)
41 \( 1 + (-0.382 + 0.923i)T^{2} \)
43 \( 1 + (0.195 + 0.980i)T^{2} \)
47 \( 1 + (-1.42 - 0.591i)T + (0.707 + 0.707i)T^{2} \)
53 \( 1 + (-0.192 - 1.95i)T + (-0.980 + 0.195i)T^{2} \)
59 \( 1 + (0.831 + 0.555i)T^{2} \)
61 \( 1 + (-1.53 + 1.26i)T + (0.195 - 0.980i)T^{2} \)
67 \( 1 + (-0.195 + 0.980i)T^{2} \)
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 \( 1 + (0.183 - 0.344i)T + (-0.555 - 0.831i)T^{2} \)
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

−9.350264389525125664447554021256, −8.498332861024472321638217686866, −7.68305960502946068500961794327, −6.93853123448907911416570976471, −6.27642338335382965564845282758, −5.52274470801818732466002294383, −4.28772375767253605765976746639, −2.76740480763780898880604927769, −2.16144952567403029596387187040, −0.959661349615277954646785574665, 1.15354620035108706012451273534, 2.42175024666545382190412833534, 3.86566418969955950609449214069, 5.06907440870230373123526592844, 5.60456870474248319389705811869, 6.31042114450035100085794282855, 7.03761495089841367111905402833, 8.214654754040759234073335373485, 8.893409258278248382987935115297, 9.548385508063944669832165982296

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