Properties

Label 2-1792-8.3-c2-0-33
Degree $2$
Conductor $1792$
Sign $-0.707 - 0.707i$
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  + 2.55·3-s + 9.86i·5-s − 2.64i·7-s − 2.47·9-s + 13.1·11-s + 5.86i·13-s + 25.2i·15-s − 0.570·17-s + 15.6·19-s − 6.75i·21-s + 16.4i·23-s − 72.3·25-s − 29.3·27-s + 29.7i·29-s + 54.8i·31-s + ⋯
L(s)  = 1  + 0.851·3-s + 1.97i·5-s − 0.377i·7-s − 0.274·9-s + 1.19·11-s + 0.451i·13-s + 1.68i·15-s − 0.0335·17-s + 0.824·19-s − 0.321i·21-s + 0.716i·23-s − 2.89·25-s − 1.08·27-s + 1.02i·29-s + 1.76i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(48.8284\)
Root analytic conductor: \(6.98773\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.343661844\)
\(L(\frac12)\) \(\approx\) \(2.343661844\)
\(L(2)\) 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 \( 1 - 2.55T + 9T^{2} \)
5 \( 1 - 9.86iT - 25T^{2} \)
11 \( 1 - 13.1T + 121T^{2} \)
13 \( 1 - 5.86iT - 169T^{2} \)
17 \( 1 + 0.570T + 289T^{2} \)
19 \( 1 - 15.6T + 361T^{2} \)
23 \( 1 - 16.4iT - 529T^{2} \)
29 \( 1 - 29.7iT - 841T^{2} \)
31 \( 1 - 54.8iT - 961T^{2} \)
37 \( 1 + 42.0iT - 1.36e3T^{2} \)
41 \( 1 + 0.773T + 1.68e3T^{2} \)
43 \( 1 - 41.7T + 1.84e3T^{2} \)
47 \( 1 + 58.4iT - 2.20e3T^{2} \)
53 \( 1 - 5.65iT - 2.80e3T^{2} \)
59 \( 1 - 42.6T + 3.48e3T^{2} \)
61 \( 1 - 95.9iT - 3.72e3T^{2} \)
67 \( 1 + 69.8T + 4.48e3T^{2} \)
71 \( 1 + 92.0iT - 5.04e3T^{2} \)
73 \( 1 + 9.97T + 5.32e3T^{2} \)
79 \( 1 - 20.1iT - 6.24e3T^{2} \)
83 \( 1 + 151.T + 6.88e3T^{2} \)
89 \( 1 + 5.79T + 7.92e3T^{2} \)
97 \( 1 - 103.T + 9.40e3T^{2} \)
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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

−9.305579782307664343685365881945, −8.747818764523230564386156650702, −7.51650420455928186489907156942, −7.16973213356394637542093136018, −6.45033925039019206953884047204, −5.52088609469338778127376827943, −3.93363938276014493003591297800, −3.43176851108153458306312006581, −2.69370365080033507493858363472, −1.60180545328914363066668434277, 0.53289431989420737226166392981, 1.55235299367950704248002676026, 2.67137496213850795347856224534, 3.90800298694595066855368511198, 4.53116252982645565950548701939, 5.56485587663140841738996882217, 6.15405267500232500777487206899, 7.63124309068873957298233510127, 8.234158965853454503170075095178, 8.758614138905524312582428940081

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