Properties

Label 2-1792-4.3-c2-0-12
Degree $2$
Conductor $1792$
Sign $1$
Analytic cond. $48.8284$
Root an. cond. $6.98773$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.22i·3-s − 6.26·5-s − 2.64i·7-s − 18.2·9-s − 9.80i·11-s + 2.41·13-s + 32.7i·15-s + 6.89·17-s + 2.77i·19-s − 13.8·21-s + 42.8i·23-s + 14.2·25-s + 48.5i·27-s − 37.3·29-s + 7.16i·31-s + ⋯
L(s)  = 1  − 1.74i·3-s − 1.25·5-s − 0.377i·7-s − 2.03·9-s − 0.891i·11-s + 0.185·13-s + 2.18i·15-s + 0.405·17-s + 0.146i·19-s − 0.658·21-s + 1.86i·23-s + 0.571·25-s + 1.79i·27-s − 1.28·29-s + 0.231i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $1$
Analytic conductor: \(48.8284\)
Root analytic conductor: \(6.98773\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1023, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5681736512\)
\(L(\frac12)\) \(\approx\) \(0.5681736512\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + 2.64iT \)
good3 \( 1 + 5.22iT - 9T^{2} \)
5 \( 1 + 6.26T + 25T^{2} \)
11 \( 1 + 9.80iT - 121T^{2} \)
13 \( 1 - 2.41T + 169T^{2} \)
17 \( 1 - 6.89T + 289T^{2} \)
19 \( 1 - 2.77iT - 361T^{2} \)
23 \( 1 - 42.8iT - 529T^{2} \)
29 \( 1 + 37.3T + 841T^{2} \)
31 \( 1 - 7.16iT - 961T^{2} \)
37 \( 1 - 0.202T + 1.36e3T^{2} \)
41 \( 1 + 63.5T + 1.68e3T^{2} \)
43 \( 1 - 35.3iT - 1.84e3T^{2} \)
47 \( 1 - 37.9iT - 2.20e3T^{2} \)
53 \( 1 - 54.6T + 2.80e3T^{2} \)
59 \( 1 + 104. iT - 3.48e3T^{2} \)
61 \( 1 - 43.7T + 3.72e3T^{2} \)
67 \( 1 - 31.1iT - 4.48e3T^{2} \)
71 \( 1 - 23.1iT - 5.04e3T^{2} \)
73 \( 1 - 69.2T + 5.32e3T^{2} \)
79 \( 1 + 19.9iT - 6.24e3T^{2} \)
83 \( 1 + 5.11iT - 6.88e3T^{2} \)
89 \( 1 - 17.9T + 7.92e3T^{2} \)
97 \( 1 - 12.4T + 9.40e3T^{2} \)
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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

−8.680351893131587984766889702557, −8.035365446639068117965341679113, −7.56041356918879045112734244966, −6.96981949232399074931342042809, −6.05960403837969805092854971920, −5.26330765601090966901905062959, −3.75042425903016715285803557131, −3.21046021983583111099318214944, −1.77003745856292977657923066914, −0.826900834475725654754600704480, 0.20150715140569849347525084978, 2.38942186367412998079099428455, 3.54336290106916114391518691677, 4.05963098076089824341304133485, 4.81257314086649610462759114973, 5.51691463242290996104079124779, 6.72894198966745542198385254458, 7.69866160142098647095539376508, 8.571915470367267902002206685244, 9.027897022813097771117230535742

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