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Properties

Label 2-3840-5.4-c1-0-29
Degree $2$
Conductor $3840$
Sign $0.113 - 0.993i$
Analytic cond. $30.6625$
Root an. cond. $5.53737$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (−2.22 − 0.254i)5-s + 2.64i·7-s − 9-s − 1.51·11-s − 3.87i·13-s + (0.254 − 2.22i)15-s + 3.31i·17-s + 7.08·19-s − 2.64·21-s − 4.82i·23-s + (4.87 + 1.12i)25-s − i·27-s + 2.18·29-s + 7.36·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.993 − 0.113i)5-s + 0.998i·7-s − 0.333·9-s − 0.456·11-s − 1.07i·13-s + (0.0656 − 0.573i)15-s + 0.803i·17-s + 1.62·19-s − 0.576·21-s − 1.00i·23-s + (0.974 + 0.225i)25-s − 0.192i·27-s + 0.405·29-s + 1.32·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.113 - 0.993i$
Analytic conductor: \(30.6625\)
Root analytic conductor: \(5.53737\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :1/2),\ 0.113 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.316185595\)
\(L(\frac12)\) \(\approx\) \(1.316185595\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (2.22 + 0.254i)T \)
good7 \( 1 - 2.64iT - 7T^{2} \)
11 \( 1 + 1.51T + 11T^{2} \)
13 \( 1 + 3.87iT - 13T^{2} \)
17 \( 1 - 3.31iT - 17T^{2} \)
19 \( 1 - 7.08T + 19T^{2} \)
23 \( 1 + 4.82iT - 23T^{2} \)
29 \( 1 - 2.18T + 29T^{2} \)
31 \( 1 - 7.36T + 31T^{2} \)
37 \( 1 + 7.87iT - 37T^{2} \)
41 \( 1 + 8.72T + 41T^{2} \)
43 \( 1 + 1.01iT - 43T^{2} \)
47 \( 1 - 7.08iT - 47T^{2} \)
53 \( 1 - 4.50iT - 53T^{2} \)
59 \( 1 - 6.79T + 59T^{2} \)
61 \( 1 + 3.60T + 61T^{2} \)
67 \( 1 + 1.01iT - 67T^{2} \)
71 \( 1 + 6.72T + 71T^{2} \)
73 \( 1 - 15.5iT - 73T^{2} \)
79 \( 1 + 7.36T + 79T^{2} \)
83 \( 1 - 7.74iT - 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 - 11.1iT - 97T^{2} \)
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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

−8.551784000690828423615098004281, −8.082212177801744463926210244586, −7.39740741392189727357127398206, −6.32328989183752214706441010938, −5.49236027954948175414371689615, −4.97242979213109711443237676594, −4.04982166565420681637417003688, −3.15991662011248931522141440397, −2.56753922600007964815330043851, −0.870992689710389334188201246538, 0.53550133565174187117667743437, 1.53016406048846478972696883978, 2.95040949165023293436060946291, 3.53565349246036946778236171750, 4.57226305720906707079325704720, 5.12846499667215184718756275760, 6.36650978303678034661058357338, 7.12503178692455892635438523806, 7.40461187083074175202730391195, 8.133956899843187879910009795031

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