Properties

Label 2-1792-1.1-c1-0-14
Degree $2$
Conductor $1792$
Sign $1$
Analytic cond. $14.3091$
Root an. cond. $3.78274$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·5-s − 7-s − 3·9-s + 2.82·11-s − 2.82·13-s − 2·17-s + 5.65·19-s + 8·23-s + 3.00·25-s − 5.65·29-s + 8·31-s − 2.82·35-s + 5.65·37-s + 6·41-s + 2.82·43-s − 8.48·45-s + 8·47-s + 49-s + 11.3·53-s + 8.00·55-s − 11.3·59-s + 2.82·61-s + 3·63-s − 8.00·65-s − 8.48·67-s + 8·71-s − 6·73-s + ⋯
L(s)  = 1  + 1.26·5-s − 0.377·7-s − 9-s + 0.852·11-s − 0.784·13-s − 0.485·17-s + 1.29·19-s + 1.66·23-s + 0.600·25-s − 1.05·29-s + 1.43·31-s − 0.478·35-s + 0.929·37-s + 0.937·41-s + 0.431·43-s − 1.26·45-s + 1.16·47-s + 0.142·49-s + 1.55·53-s + 1.07·55-s − 1.47·59-s + 0.362·61-s + 0.377·63-s − 0.992·65-s − 1.03·67-s + 0.949·71-s − 0.702·73-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.070289289\)
\(L(\frac12)\) \(\approx\) \(2.070289289\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good3 \( 1 + 3T^{2} \)
5 \( 1 - 2.82T + 5T^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 + 5.65T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 5.65T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 2.82T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 2.82T + 61T^{2} \)
67 \( 1 + 8.48T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 5.65T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−9.283207291184141466584345986319, −8.849715706945957713738810475150, −7.58342662965889560029930902122, −6.82202198869697479163613700476, −5.94105328764709052778105350747, −5.45851803601920151136071006506, −4.41284954827906460242568726079, −3.06394302889213970363524277298, −2.41233067412755602077374319604, −1.02142960226701437714894546397, 1.02142960226701437714894546397, 2.41233067412755602077374319604, 3.06394302889213970363524277298, 4.41284954827906460242568726079, 5.45851803601920151136071006506, 5.94105328764709052778105350747, 6.82202198869697479163613700476, 7.58342662965889560029930902122, 8.849715706945957713738810475150, 9.283207291184141466584345986319

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