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Properties

Label 2-3840-3840.1469-c0-0-0
Degree $2$
Conductor $3840$
Sign $0.359 + 0.932i$
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  + (0.857 − 0.514i)2-s + (−0.941 + 0.336i)3-s + (0.471 − 0.881i)4-s + (0.803 − 0.595i)5-s + (−0.634 + 0.773i)6-s + (−0.0490 − 0.998i)8-s + (0.773 − 0.634i)9-s + (0.382 − 0.923i)10-s + (−0.146 + 0.989i)12-s + (−0.555 + 0.831i)15-s + (−0.555 − 0.831i)16-s + (1.09 + 1.64i)17-s + (0.336 − 0.941i)18-s + (0.0504 + 0.0841i)19-s + (−0.146 − 0.989i)20-s + ⋯
L(s)  = 1  + (0.857 − 0.514i)2-s + (−0.941 + 0.336i)3-s + (0.471 − 0.881i)4-s + (0.803 − 0.595i)5-s + (−0.634 + 0.773i)6-s + (−0.0490 − 0.998i)8-s + (0.773 − 0.634i)9-s + (0.382 − 0.923i)10-s + (−0.146 + 0.989i)12-s + (−0.555 + 0.831i)15-s + (−0.555 − 0.831i)16-s + (1.09 + 1.64i)17-s + (0.336 − 0.941i)18-s + (0.0504 + 0.0841i)19-s + (−0.146 − 0.989i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.359 + 0.932i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ 0.359 + 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.955298891\)
\(L(\frac12)\) \(\approx\) \(1.955298891\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.857 + 0.514i)T \)
3 \( 1 + (0.941 - 0.336i)T \)
5 \( 1 + (-0.803 + 0.595i)T \)
good7 \( 1 + (-0.980 + 0.195i)T^{2} \)
11 \( 1 + (0.0980 + 0.995i)T^{2} \)
13 \( 1 + (0.956 - 0.290i)T^{2} \)
17 \( 1 + (-1.09 - 1.64i)T + (-0.382 + 0.923i)T^{2} \)
19 \( 1 + (-0.0504 - 0.0841i)T + (-0.471 + 0.881i)T^{2} \)
23 \( 1 + (-1.71 - 0.914i)T + (0.555 + 0.831i)T^{2} \)
29 \( 1 + (-0.995 - 0.0980i)T^{2} \)
31 \( 1 + (0.674 + 1.62i)T + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (-0.881 + 0.471i)T^{2} \)
41 \( 1 + (0.831 - 0.555i)T^{2} \)
43 \( 1 + (0.773 + 0.634i)T^{2} \)
47 \( 1 + (1.77 + 0.352i)T + (0.923 + 0.382i)T^{2} \)
53 \( 1 + (-0.195 + 0.00961i)T + (0.995 - 0.0980i)T^{2} \)
59 \( 1 + (0.956 + 0.290i)T^{2} \)
61 \( 1 + (1.21 + 0.574i)T + (0.634 + 0.773i)T^{2} \)
67 \( 1 + (0.634 + 0.773i)T^{2} \)
71 \( 1 + (0.195 + 0.980i)T^{2} \)
73 \( 1 + (-0.980 - 0.195i)T^{2} \)
79 \( 1 + (-0.216 - 1.08i)T + (-0.923 + 0.382i)T^{2} \)
83 \( 1 + (1.49 + 0.375i)T + (0.881 + 0.471i)T^{2} \)
89 \( 1 + (-0.555 + 0.831i)T^{2} \)
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

−8.754976334151284026313872981123, −7.59610945463091127382222511223, −6.68633134827692936170181548434, −5.89347611929800609325375905935, −5.56348808787034171734903330177, −4.85266427698434991760666515872, −4.01763698442994564838030174297, −3.24506776499063633577370946611, −1.85587927076272745812810865765, −1.11080994432832434746151460933, 1.38871044047477092385636699953, 2.67655413588398577722202642395, 3.26797546154278265673661624711, 4.70108375886782294619843086255, 5.14532108604631468462947809723, 5.77179392305812511165414418044, 6.61573008516313996191927616692, 7.07725875475548308791239990399, 7.56883017986990507577668825032, 8.731382549967848477030247734676

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