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Properties

Label 2-3840-3840.389-c0-0-1
Degree $2$
Conductor $3840$
Sign $0.985 + 0.170i$
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  + (0.427 + 0.903i)2-s + (−0.989 − 0.146i)3-s + (−0.634 + 0.773i)4-s + (0.0490 − 0.998i)5-s + (−0.290 − 0.956i)6-s + (−0.970 − 0.242i)8-s + (0.956 + 0.290i)9-s + (0.923 − 0.382i)10-s + (0.740 − 0.671i)12-s + (−0.195 + 0.980i)15-s + (−0.195 − 0.980i)16-s + (0.262 + 1.31i)17-s + (0.146 + 0.989i)18-s + (1.75 − 0.829i)19-s + (0.740 + 0.671i)20-s + ⋯
L(s)  = 1  + (0.427 + 0.903i)2-s + (−0.989 − 0.146i)3-s + (−0.634 + 0.773i)4-s + (0.0490 − 0.998i)5-s + (−0.290 − 0.956i)6-s + (−0.970 − 0.242i)8-s + (0.956 + 0.290i)9-s + (0.923 − 0.382i)10-s + (0.740 − 0.671i)12-s + (−0.195 + 0.980i)15-s + (−0.195 − 0.980i)16-s + (0.262 + 1.31i)17-s + (0.146 + 0.989i)18-s + (1.75 − 0.829i)19-s + (0.740 + 0.671i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.985 + 0.170i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ 0.985 + 0.170i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9383650069\)
\(L(\frac12)\) \(\approx\) \(0.9383650069\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.427 - 0.903i)T \)
3 \( 1 + (0.989 + 0.146i)T \)
5 \( 1 + (-0.0490 + 0.998i)T \)
good7 \( 1 + (-0.555 - 0.831i)T^{2} \)
11 \( 1 + (-0.471 + 0.881i)T^{2} \)
13 \( 1 + (-0.0980 - 0.995i)T^{2} \)
17 \( 1 + (-0.262 - 1.31i)T + (-0.923 + 0.382i)T^{2} \)
19 \( 1 + (-1.75 + 0.829i)T + (0.634 - 0.773i)T^{2} \)
23 \( 1 + (1.45 + 1.19i)T + (0.195 + 0.980i)T^{2} \)
29 \( 1 + (0.881 - 0.471i)T^{2} \)
31 \( 1 + (1.42 + 0.591i)T + (0.707 + 0.707i)T^{2} \)
37 \( 1 + (-0.773 + 0.634i)T^{2} \)
41 \( 1 + (-0.980 + 0.195i)T^{2} \)
43 \( 1 + (0.956 - 0.290i)T^{2} \)
47 \( 1 + (-0.892 + 1.33i)T + (-0.382 - 0.923i)T^{2} \)
53 \( 1 + (-0.229 + 0.914i)T + (-0.881 - 0.471i)T^{2} \)
59 \( 1 + (-0.0980 + 0.995i)T^{2} \)
61 \( 1 + (-1.37 + 1.02i)T + (0.290 - 0.956i)T^{2} \)
67 \( 1 + (0.290 - 0.956i)T^{2} \)
71 \( 1 + (0.831 - 0.555i)T^{2} \)
73 \( 1 + (-0.555 + 0.831i)T^{2} \)
79 \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \)
83 \( 1 + (-1.80 - 0.644i)T + (0.773 + 0.634i)T^{2} \)
89 \( 1 + (-0.195 + 0.980i)T^{2} \)
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

−8.398611875943155367464015488196, −7.82240097704900670193064363328, −7.08612154179209734106814006395, −6.27109701884509046116041499816, −5.56966928515203017170705820946, −5.19515192857245794883420865166, −4.25809312259984784000306285751, −3.69751391472323830269804492784, −2.03393420511932555783319287322, −0.61147947821452035314328191052, 1.16285277485286140219909698314, 2.27869059867036731296473503585, 3.41781498944737130216091328267, 3.84160499140549784037041085378, 5.03888437417603306087677596678, 5.61057433082015446925233148806, 6.12965522030316656914188827598, 7.27522381888904314020624886767, 7.59660658881016502308636575297, 9.126935336104281903295288905453

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