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Properties

Label 2-1792-16.13-c1-0-16
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  + (2.04 + 2.04i)3-s + (0.701 − 0.701i)5-s + i·7-s + 5.33i·9-s + (−2.41 + 2.41i)11-s + (1.96 + 1.96i)13-s + 2.86·15-s − 6.93·17-s + (1.38 + 1.38i)19-s + (−2.04 + 2.04i)21-s + 2.05i·23-s + 4.01i·25-s + (−4.76 + 4.76i)27-s + (−5.34 − 5.34i)29-s + 5.23·31-s + ⋯
L(s)  = 1  + (1.17 + 1.17i)3-s + (0.313 − 0.313i)5-s + 0.377i·7-s + 1.77i·9-s + (−0.729 + 0.729i)11-s + (0.543 + 0.543i)13-s + 0.739·15-s − 1.68·17-s + (0.318 + 0.318i)19-s + (−0.445 + 0.445i)21-s + 0.428i·23-s + 0.803i·25-s + (−0.917 + 0.917i)27-s + (−0.992 − 0.992i)29-s + 0.940·31-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: $\chi_{1792} (1345, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ -0.608 - 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.351475477\)
\(L(\frac12)\) \(\approx\) \(2.351475477\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.04 - 2.04i)T + 3iT^{2} \)
5 \( 1 + (-0.701 + 0.701i)T - 5iT^{2} \)
11 \( 1 + (2.41 - 2.41i)T - 11iT^{2} \)
13 \( 1 + (-1.96 - 1.96i)T + 13iT^{2} \)
17 \( 1 + 6.93T + 17T^{2} \)
19 \( 1 + (-1.38 - 1.38i)T + 19iT^{2} \)
23 \( 1 - 2.05iT - 23T^{2} \)
29 \( 1 + (5.34 + 5.34i)T + 29iT^{2} \)
31 \( 1 - 5.23T + 31T^{2} \)
37 \( 1 + (-6.58 + 6.58i)T - 37iT^{2} \)
41 \( 1 - 0.949iT - 41T^{2} \)
43 \( 1 + (5.95 - 5.95i)T - 43iT^{2} \)
47 \( 1 - 4.64T + 47T^{2} \)
53 \( 1 + (7.24 - 7.24i)T - 53iT^{2} \)
59 \( 1 + (-8.58 + 8.58i)T - 59iT^{2} \)
61 \( 1 + (2.81 + 2.81i)T + 61iT^{2} \)
67 \( 1 + (-9.07 - 9.07i)T + 67iT^{2} \)
71 \( 1 - 3.60iT - 71T^{2} \)
73 \( 1 - 6.53iT - 73T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 + (4.49 + 4.49i)T + 83iT^{2} \)
89 \( 1 - 0.428iT - 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−9.536482634283242210116823477377, −8.937782900306830229125633253208, −8.222774385994132706687384609022, −7.46616298045274180910548760283, −6.28340695553184249640821134496, −5.24355697364103319886479073894, −4.47463825302819622006960451255, −3.80823171869165267494247236579, −2.64815161828023949214503061975, −1.94401294284206074848818792252, 0.71161403809924926764023239420, 2.05424133742061141742400610342, 2.78735710526899824269712873773, 3.59993210684488200210549355520, 4.88185496162507812753243698409, 6.16441490557418325371594309385, 6.69632466333263107347646045573, 7.48468898422016539833243388071, 8.336470794037563170920649026577, 8.628431907294836435333095255201

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