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Properties

Label 2-3840-60.47-c0-0-4
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: $\chi_{3840} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.832865119\)
\(L(\frac12)\) \(\approx\) \(1.832865119\)
\(L(1)\) 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

−8.673338681981941281078947636565, −8.031868486856094262435079902027, −6.83817255090561042990971532186, −6.39133246419076900514004831921, −5.86020453396055703376354290910, −4.78946188920796628211610031668, −3.74584049178384344512126996150, −2.89445357827259924065900657371, −2.05805311428649780684885372400, −1.09431749464271797304538208354, 1.41932606126051930608969199379, 2.64979522737496586722146934879, 3.38785875734239922650032173048, 3.98714795996220034043922692071, 4.83033071393344860270272209406, 6.01789508185612745234499432321, 6.63418643435422962540157860305, 7.17052606878659564115083949244, 8.119828494596361960718871413657, 9.014360203510481917802148359923

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