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Properties

Label 2-3840-3840.869-c0-0-0
Degree $2$
Conductor $3840$
Sign $-0.817 + 0.575i$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
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  + (−0.671 + 0.740i)2-s + (−0.970 + 0.242i)3-s + (−0.0980 − 0.995i)4-s + (0.903 + 0.427i)5-s + (0.471 − 0.881i)6-s + (0.803 + 0.595i)8-s + (0.881 − 0.471i)9-s + (−0.923 + 0.382i)10-s + (0.336 + 0.941i)12-s + (−0.980 − 0.195i)15-s + (−0.980 + 0.195i)16-s + (−1.84 + 0.367i)17-s + (−0.242 + 0.970i)18-s + (−1.19 − 1.07i)19-s + (0.336 − 0.941i)20-s + ⋯
L(s)  = 1  + (−0.671 + 0.740i)2-s + (−0.970 + 0.242i)3-s + (−0.0980 − 0.995i)4-s + (0.903 + 0.427i)5-s + (0.471 − 0.881i)6-s + (0.803 + 0.595i)8-s + (0.881 − 0.471i)9-s + (−0.923 + 0.382i)10-s + (0.336 + 0.941i)12-s + (−0.980 − 0.195i)15-s + (−0.980 + 0.195i)16-s + (−1.84 + 0.367i)17-s + (−0.242 + 0.970i)18-s + (−1.19 − 1.07i)19-s + (0.336 − 0.941i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $-0.817 + 0.575i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (869, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ -0.817 + 0.575i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1424509433\)
\(L(\frac12)\) \(\approx\) \(0.1424509433\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.671 - 0.740i)T \)
3 \( 1 + (0.970 - 0.242i)T \)
5 \( 1 + (-0.903 - 0.427i)T \)
good7 \( 1 + (0.831 - 0.555i)T^{2} \)
11 \( 1 + (-0.956 + 0.290i)T^{2} \)
13 \( 1 + (0.773 + 0.634i)T^{2} \)
17 \( 1 + (1.84 - 0.367i)T + (0.923 - 0.382i)T^{2} \)
19 \( 1 + (1.19 + 1.07i)T + (0.0980 + 0.995i)T^{2} \)
23 \( 1 + (-0.0976 + 0.00961i)T + (0.980 - 0.195i)T^{2} \)
29 \( 1 + (0.290 - 0.956i)T^{2} \)
31 \( 1 + (1.83 + 0.761i)T + (0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.995 + 0.0980i)T^{2} \)
41 \( 1 + (0.195 + 0.980i)T^{2} \)
43 \( 1 + (0.881 + 0.471i)T^{2} \)
47 \( 1 + (0.854 + 0.571i)T + (0.382 + 0.923i)T^{2} \)
53 \( 1 + (1.14 - 1.53i)T + (-0.290 - 0.956i)T^{2} \)
59 \( 1 + (0.773 - 0.634i)T^{2} \)
61 \( 1 + (1.01 - 1.69i)T + (-0.471 - 0.881i)T^{2} \)
67 \( 1 + (-0.471 - 0.881i)T^{2} \)
71 \( 1 + (0.555 + 0.831i)T^{2} \)
73 \( 1 + (0.831 + 0.555i)T^{2} \)
79 \( 1 + (-1.08 - 1.63i)T + (-0.382 + 0.923i)T^{2} \)
83 \( 1 + (-0.0865 - 1.76i)T + (-0.995 + 0.0980i)T^{2} \)
89 \( 1 + (-0.980 - 0.195i)T^{2} \)
97 \( 1 + (0.707 + 0.707i)T^{2} \)
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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

−9.260973461148740063641297070907, −8.507004248252627886406086516197, −7.36384901960965819692162905457, −6.71945514595788546579647653298, −6.32115128780130303891574021489, −5.62390137675483920318172900891, −4.81488167448992620520734447184, −4.13598183283144850324068349656, −2.44878897865962033335765044525, −1.57925702375113161428320586659, 0.10744643447512551544660715270, 1.75700286317020125882911487929, 1.99356735836252688916931039032, 3.47396463888914625153956910055, 4.56493476175997984397148174198, 5.03813901566611986484789111302, 6.26636089548918448616882249569, 6.58924534445694508695747187957, 7.55962423104940999010931946183, 8.420706638330500183966786927982

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