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Properties

Label 2-3840-60.23-c0-0-2
Degree $2$
Conductor $3840$
Sign $0.525 - 0.850i$
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  + (−0.707 − 0.707i)3-s + (0.707 + 0.707i)5-s + (1 + i)7-s + 1.00i·9-s − 1.41·11-s − 1.00i·15-s − 1.41i·21-s + 1.00i·25-s + (0.707 − 0.707i)27-s + 1.41·29-s + (1.00 + 1.00i)33-s + 1.41i·35-s + (−0.707 + 0.707i)45-s + i·49-s + (−1.41 + 1.41i)53-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (0.707 + 0.707i)5-s + (1 + i)7-s + 1.00i·9-s − 1.41·11-s − 1.00i·15-s − 1.41i·21-s + 1.00i·25-s + (0.707 − 0.707i)27-s + 1.41·29-s + (1.00 + 1.00i)33-s + 1.41i·35-s + (−0.707 + 0.707i)45-s + i·49-s + (−1.41 + 1.41i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (2303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.110616639\)
\(L(\frac12)\) \(\approx\) \(1.110616639\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (-1 - i)T + iT^{2} \)
11 \( 1 + 1.41T + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
59 \( 1 - 1.41iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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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

−8.544120189816590650244304592459, −7.994555034297999380594150413705, −7.31462254367513006765533777099, −6.47827549846626627723040818746, −5.75688035160667225561476415839, −5.27838361210119744488556777052, −4.58510579423479366442850113138, −2.83864395593497002629844771162, −2.39244170395646906001462564606, −1.43038480121968904847828569886, 0.71277772928308248261272854590, 1.87372507299942308481555422381, 3.14355007365974407104058942416, 4.30269483492614172324815923308, 4.88132138952124805084884098702, 5.27275563475136631108103883676, 6.18646760092108592139629029278, 6.99542514875562879885153717616, 8.051502580584650379817039035481, 8.382003972259418133476636436080

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