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Properties

Label 2-3840-5.4-c1-0-83
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

\(L(1)\) \(\approx\) \(1.788451599\)
\(L(\frac12)\) \(\approx\) \(1.788451599\)
\(L(\frac{3}{2})\) 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

−8.455864487727436711851346508784, −7.56172908686335416904329802340, −6.78514090639766986271880235329, −6.03319172904469780165323351168, −5.29627426419826602224692148301, −4.53375524790354687852445435002, −3.72681321116512637269191963873, −2.82058128837826150640843903514, −1.73201632363459262191520553361, −0.49055309046477534559024994669, 1.44414020674408031574681895177, 2.13511813968228161038389543833, 2.84257516171418857247867198897, 4.17184936322275478500191607435, 4.95825676988224175967836921198, 6.00119279194721124176217112155, 6.42479720819244769413874106255, 6.81908603945215911336073989470, 8.191946307255354258125248899782, 8.636224807334552295162765987566

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