Properties

Label 2-1792-16.5-c1-0-41
Degree $2$
Conductor $1792$
Sign $0.608 + 0.793i$
Analytic cond. $14.3091$
Root an. cond. $3.78274$
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  + (2.41 − 2.41i)3-s + (2.54 + 2.54i)5-s + i·7-s − 8.70i·9-s + (−0.764 − 0.764i)11-s + (1.26 − 1.26i)13-s + 12.2·15-s + 5.65·17-s + (−0.0445 + 0.0445i)19-s + (2.41 + 2.41i)21-s − 1.46i·23-s + 7.91i·25-s + (−13.8 − 13.8i)27-s + (−3.56 + 3.56i)29-s − 4.75·31-s + ⋯
L(s)  = 1  + (1.39 − 1.39i)3-s + (1.13 + 1.13i)5-s + 0.377i·7-s − 2.90i·9-s + (−0.230 − 0.230i)11-s + (0.351 − 0.351i)13-s + 3.17·15-s + 1.37·17-s + (−0.0102 + 0.0102i)19-s + (0.527 + 0.527i)21-s − 0.305i·23-s + 1.58i·25-s + (−2.65 − 2.65i)27-s + (−0.662 + 0.662i)29-s − 0.853·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.608 + 0.793i$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ 0.608 + 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.546223442\)
\(L(\frac12)\) \(\approx\) \(3.546223442\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - iT \)
good3 \( 1 + (-2.41 + 2.41i)T - 3iT^{2} \)
5 \( 1 + (-2.54 - 2.54i)T + 5iT^{2} \)
11 \( 1 + (0.764 + 0.764i)T + 11iT^{2} \)
13 \( 1 + (-1.26 + 1.26i)T - 13iT^{2} \)
17 \( 1 - 5.65T + 17T^{2} \)
19 \( 1 + (0.0445 - 0.0445i)T - 19iT^{2} \)
23 \( 1 + 1.46iT - 23T^{2} \)
29 \( 1 + (3.56 - 3.56i)T - 29iT^{2} \)
31 \( 1 + 4.75T + 31T^{2} \)
37 \( 1 + (-5.09 - 5.09i)T + 37iT^{2} \)
41 \( 1 - 7.50iT - 41T^{2} \)
43 \( 1 + (3.22 + 3.22i)T + 43iT^{2} \)
47 \( 1 - 1.52T + 47T^{2} \)
53 \( 1 + (4.66 + 4.66i)T + 53iT^{2} \)
59 \( 1 + (-5.38 - 5.38i)T + 59iT^{2} \)
61 \( 1 + (-6.80 + 6.80i)T - 61iT^{2} \)
67 \( 1 + (4.92 - 4.92i)T - 67iT^{2} \)
71 \( 1 + 6.19iT - 71T^{2} \)
73 \( 1 + 8.59iT - 73T^{2} \)
79 \( 1 + 7.84T + 79T^{2} \)
83 \( 1 + (-7.43 + 7.43i)T - 83iT^{2} \)
89 \( 1 - 9.32iT - 89T^{2} \)
97 \( 1 - 0.485T + 97T^{2} \)
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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

−9.136169712720462317802173013868, −8.211972598712816684963686160630, −7.64910874446617739741783881151, −6.82201834128959239570166869903, −6.22075821792148183856255572946, −5.50155742579546223738728438106, −3.49650289489358980301531696290, −3.00216051065976483223267421149, −2.18917315472997920411263295558, −1.30631590201103069961040754490, 1.55637878589169235210736494083, 2.48820749219019840950204638049, 3.65237926572438844411237821619, 4.30686024023124319659380086002, 5.25556907236541639563323246027, 5.71682773549374282549057761531, 7.38771029718275050504019307348, 8.097715125293779897485086663869, 8.857981051923589402924193904953, 9.429371755160350842399753397308

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