Properties

Label 2-3840-480.149-c0-0-2
Degree $2$
Conductor $3840$
Sign $0.195 + 0.980i$
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

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

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.195 + 0.980i$
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: \((2,\ 3840,\ (\ :0),\ 0.195 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4290772590\)
\(L(\frac12)\) \(\approx\) \(0.4290772590\)
\(L(1)\) 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

−8.565281141776478860160783225745, −7.46446238096068360728587044963, −6.99043545873165237413718947610, −6.37735651383471689990163521201, −5.50030035548003184541349972186, −4.76398116278549248463651634622, −3.93624417229941060086304764180, −3.13618023440832369682196949354, −2.03324661328565186240204089561, −0.28780490475417867853645958928, 1.27083398269622641323926772097, 2.06797710748769756728092611893, 3.74179661941220416939354769724, 4.27481345635356732150143947334, 5.19294093708621070422109194480, 5.86744281125613321433721395568, 6.43009989875747999478729950173, 7.47554319625722242896424193145, 8.037204238832448877373837570036, 8.630237730020813806907480962631

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