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Properties

Label 2-1920-1920.269-c0-0-1
Degree $2$
Conductor $1920$
Sign $-0.903 + 0.427i$
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  + (−0.773 + 0.634i)2-s + (−0.290 − 0.956i)3-s + (0.195 − 0.980i)4-s + (−0.0980 − 0.995i)5-s + (0.831 + 0.555i)6-s + (0.471 + 0.881i)8-s + (−0.831 + 0.555i)9-s + (0.707 + 0.707i)10-s + (−0.995 + 0.0980i)12-s + (−0.923 + 0.382i)15-s + (−0.923 − 0.382i)16-s + (−1.83 − 0.761i)17-s + (0.290 − 0.956i)18-s + (0.728 − 0.598i)19-s + (−0.995 − 0.0980i)20-s + ⋯
L(s)  = 1  + (−0.773 + 0.634i)2-s + (−0.290 − 0.956i)3-s + (0.195 − 0.980i)4-s + (−0.0980 − 0.995i)5-s + (0.831 + 0.555i)6-s + (0.471 + 0.881i)8-s + (−0.831 + 0.555i)9-s + (0.707 + 0.707i)10-s + (−0.995 + 0.0980i)12-s + (−0.923 + 0.382i)15-s + (−0.923 − 0.382i)16-s + (−1.83 − 0.761i)17-s + (0.290 − 0.956i)18-s + (0.728 − 0.598i)19-s + (−0.995 − 0.0980i)20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.903 + 0.427i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :0),\ -0.903 + 0.427i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3951090813\)
\(L(\frac12)\) \(\approx\) \(0.3951090813\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.773 - 0.634i)T \)
3 \( 1 + (0.290 + 0.956i)T \)
5 \( 1 + (0.0980 + 0.995i)T \)
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 \( 1 + (1.83 + 0.761i)T + (0.707 + 0.707i)T^{2} \)
19 \( 1 + (-0.728 + 0.598i)T + (0.195 - 0.980i)T^{2} \)
23 \( 1 + (-0.247 + 1.24i)T + (-0.923 - 0.382i)T^{2} \)
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 \( 1 + (0.732 - 1.76i)T + (-0.707 - 0.707i)T^{2} \)
53 \( 1 + (0.523 + 0.979i)T + (-0.555 + 0.831i)T^{2} \)
59 \( 1 + (-0.980 + 0.195i)T^{2} \)
61 \( 1 + (1.68 - 0.512i)T + (0.831 - 0.555i)T^{2} \)
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 \( 1 + (-1.05 - 1.28i)T + (-0.195 + 0.980i)T^{2} \)
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

−8.949066091555232966350850989499, −8.264658962737004905247091148324, −7.52262936746341036829432220575, −6.75381883557492107933727713257, −6.16863414084870859714936532233, −5.06661836045069982570930027994, −4.64388229389287327390593181022, −2.65467307769339788983749304417, −1.60196133330961280124686693963, −0.37702594486190877847593637267, 1.89466460395169863768179731221, 3.05479386196962167649078366622, 3.71625434849378105863597073827, 4.56298517944447498284716628819, 5.85494777670931655699855485748, 6.65326006610678720637324104497, 7.50876796706630421147745981817, 8.372704099063283796269143651610, 9.215691336993971830829930474754, 9.758915109912552667517419381126

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