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Properties

Label 2-3840-60.23-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(1.741534615\)
\(L(\frac12)\) \(\approx\) \(1.741534615\)
\(L(1)\) 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

−8.863649167888022106396407380010, −8.206617881562068422659378961936, −7.65362113200304438856759755389, −6.64230101451984365577995713263, −5.41938169501641508531153715317, −5.04948271793412683374752771223, −4.03455751070035986114355866067, −3.66447215762672565797728999057, −2.33957446648511860572066165083, −1.48313136041689213613880527136, 1.04613862763578734483974676744, 1.95250287519394381547673118867, 3.11721939483100684136046239946, 3.98436900901085584906199473858, 4.30506448921932296467846255592, 5.75922096264173737966463932751, 6.66374432444458043572481100111, 7.26529720919933589567889661864, 7.62260017799145325102984773988, 8.429105549043784317370926059512

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