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Properties

Label 2-1920-1920.389-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: $\chi_{1920} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :0),\ -0.427 + 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.342058792\)
\(L(\frac12)\) \(\approx\) \(1.342058792\)
\(L(1)\) 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

−9.548371537106066114425770405501, −8.577856217622875057706214584250, −7.18307688617482357398860830524, −6.48400335596443868232104254518, −5.87382609962848777705990551935, −5.00087583960738030352609225256, −4.50519715114672908358362805117, −3.02953465010534006398435480071, −2.13259106406426040867456880306, −0.984643075519604886101241121566, 1.62379898251794784201869436310, 3.18506696337845341771873356682, 4.11281964544669360056335574104, 5.13111038920741816657284227419, 5.64740147579137892277136501945, 6.18818644940989492691636294319, 7.08738827077406799728514403914, 7.80611658486697960200080713862, 8.914458830684251264390383865872, 9.710063343198948032123117765714

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