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Properties

Label 2-1920-1920.749-c0-0-1
Degree $2$
Conductor $1920$
Sign $0.336 - 0.941i$
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  + (−0.0980 + 0.995i)2-s + (0.881 + 0.471i)3-s + (−0.980 − 0.195i)4-s + (0.634 − 0.773i)5-s + (−0.555 + 0.831i)6-s + (0.290 − 0.956i)8-s + (0.555 + 0.831i)9-s + (0.707 + 0.707i)10-s + (−0.773 − 0.634i)12-s + (0.923 − 0.382i)15-s + (0.923 + 0.382i)16-s + (1.42 + 0.591i)17-s + (−0.881 + 0.471i)18-s + (0.0569 − 0.577i)19-s + (−0.773 + 0.634i)20-s + ⋯
L(s)  = 1  + (−0.0980 + 0.995i)2-s + (0.881 + 0.471i)3-s + (−0.980 − 0.195i)4-s + (0.634 − 0.773i)5-s + (−0.555 + 0.831i)6-s + (0.290 − 0.956i)8-s + (0.555 + 0.831i)9-s + (0.707 + 0.707i)10-s + (−0.773 − 0.634i)12-s + (0.923 − 0.382i)15-s + (0.923 + 0.382i)16-s + (1.42 + 0.591i)17-s + (−0.881 + 0.471i)18-s + (0.0569 − 0.577i)19-s + (−0.773 + 0.634i)20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.336 - 0.941i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :0),\ 0.336 - 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.534153825\)
\(L(\frac12)\) \(\approx\) \(1.534153825\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0980 - 0.995i)T \)
3 \( 1 + (-0.881 - 0.471i)T \)
5 \( 1 + (-0.634 + 0.773i)T \)
good7 \( 1 + (-0.382 - 0.923i)T^{2} \)
11 \( 1 + (-0.831 - 0.555i)T^{2} \)
13 \( 1 + (-0.195 + 0.980i)T^{2} \)
17 \( 1 + (-1.42 - 0.591i)T + (0.707 + 0.707i)T^{2} \)
19 \( 1 + (-0.0569 + 0.577i)T + (-0.980 - 0.195i)T^{2} \)
23 \( 1 + (1.95 + 0.388i)T + (0.923 + 0.382i)T^{2} \)
29 \( 1 + (0.831 - 0.555i)T^{2} \)
31 \( 1 + (-1.38 + 1.38i)T - iT^{2} \)
37 \( 1 + (-0.980 + 0.195i)T^{2} \)
41 \( 1 + (0.923 + 0.382i)T^{2} \)
43 \( 1 + (-0.555 + 0.831i)T^{2} \)
47 \( 1 + (0.360 - 0.871i)T + (-0.707 - 0.707i)T^{2} \)
53 \( 1 + (0.482 - 1.59i)T + (-0.831 - 0.555i)T^{2} \)
59 \( 1 + (-0.195 - 0.980i)T^{2} \)
61 \( 1 + (0.902 - 1.68i)T + (-0.555 - 0.831i)T^{2} \)
67 \( 1 + (0.555 + 0.831i)T^{2} \)
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 \( 1 + (1.10 + 0.108i)T + (0.980 + 0.195i)T^{2} \)
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

−9.432368360249981925513530863793, −8.673267742896213492070808892650, −7.999172403049501871719324969754, −7.54637381408742789366181124709, −6.13678851011703757313682903722, −5.73827498125706631555425105777, −4.55493480367973433229476209756, −4.15234568086816841321835889626, −2.80814472448383068038664529769, −1.41310141165737786345959094492, 1.42449204432030177266256087033, 2.27584919086735991163549378568, 3.22035348753197686404053804461, 3.75290717227223949467603968902, 5.11282173526076216748669919564, 6.06628392049662501716518513920, 7.03306124024960230859373740602, 7.969510256386482598574130056170, 8.407055011474932138125602031238, 9.613837840037194537662451454947

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