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Properties

Label 2-3840-480.149-c0-0-1
Degree $2$
Conductor $3840$
Sign $0.980 - 0.195i$
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  + (0.923 − 0.382i)3-s + (−0.382 + 0.923i)5-s + (0.707 − 0.707i)9-s + i·15-s − 0.765i·17-s + (0.707 + 1.70i)19-s + (0.541 − 0.541i)23-s + (−0.707 − 0.707i)25-s + (0.382 − 0.923i)27-s + 1.41·31-s + (0.382 + 0.923i)45-s + 1.84i·47-s + i·49-s + (−0.292 − 0.707i)51-s + (−1.30 − 0.541i)53-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s + (−0.382 + 0.923i)5-s + (0.707 − 0.707i)9-s + i·15-s − 0.765i·17-s + (0.707 + 1.70i)19-s + (0.541 − 0.541i)23-s + (−0.707 − 0.707i)25-s + (0.382 − 0.923i)27-s + 1.41·31-s + (0.382 + 0.923i)45-s + 1.84i·47-s + i·49-s + (−0.292 − 0.707i)51-s + (−1.30 − 0.541i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.980 - 0.195i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ 0.980 - 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.764370650\)
\(L(\frac12)\) \(\approx\) \(1.764370650\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.923 + 0.382i)T \)
5 \( 1 + (0.382 - 0.923i)T \)
good7 \( 1 - iT^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + (0.707 - 0.707i)T^{2} \)
17 \( 1 + 0.765iT - T^{2} \)
19 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
29 \( 1 + (-0.707 + 0.707i)T^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 + (0.707 + 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 - 1.84iT - T^{2} \)
53 \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
67 \( 1 + (-0.707 + 0.707i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 - T^{2} \)
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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

−8.403653907861075755278745559228, −7.977948212930455207572685689188, −7.33310730708212407325390688140, −6.62834468451190631112055552367, −5.97803541877419252743115817074, −4.74555221615205923390165580638, −3.85852165487777943789192617072, −3.10030632085968086679092660781, −2.49936280312918972225167951590, −1.26293220235279327425848229223, 1.10052692224474242660944585280, 2.27997104585095315070104908077, 3.25651980777871846865705135643, 4.01295954558099934925101962001, 4.84375857165629016296770726443, 5.31517192474195242491354077507, 6.60485847737682741703174922792, 7.36552083281959682716021480399, 8.080719285542108757186228825074, 8.695027378057236585961699934325

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