Properties

Label 2-1792-1.1-c1-0-6
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  − 3.02·3-s + 1.69·5-s − 7-s + 6.12·9-s − 1.32·11-s + 1.69·13-s − 5.12·15-s + 2·17-s − 3.02·19-s + 3.02·21-s + 5.12·23-s − 2.12·25-s − 9.43·27-s − 6.04·29-s + 10.2·31-s + 4·33-s − 1.69·35-s − 6.04·37-s − 5.12·39-s − 4.24·41-s − 1.32·43-s + 10.3·45-s + 49-s − 6.04·51-s + 2.64·53-s − 2.24·55-s + 9.12·57-s + ⋯
L(s)  = 1  − 1.74·3-s + 0.758·5-s − 0.377·7-s + 2.04·9-s − 0.399·11-s + 0.470·13-s − 1.32·15-s + 0.485·17-s − 0.692·19-s + 0.659·21-s + 1.06·23-s − 0.424·25-s − 1.81·27-s − 1.12·29-s + 1.84·31-s + 0.696·33-s − 0.286·35-s − 0.993·37-s − 0.820·39-s − 0.663·41-s − 0.201·43-s + 1.54·45-s + 0.142·49-s − 0.845·51-s + 0.363·53-s − 0.302·55-s + 1.20·57-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

\(L(1)\) \(\approx\) \(0.9415185585\)
\(L(\frac12)\) \(\approx\) \(0.9415185585\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.02T + 3T^{2} \)
5 \( 1 - 1.69T + 5T^{2} \)
11 \( 1 + 1.32T + 11T^{2} \)
13 \( 1 - 1.69T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 3.02T + 19T^{2} \)
23 \( 1 - 5.12T + 23T^{2} \)
29 \( 1 + 6.04T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + 6.04T + 37T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + 1.32T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 2.64T + 53T^{2} \)
59 \( 1 + 0.371T + 59T^{2} \)
61 \( 1 - 1.69T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 5.66T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−9.571906689743621188286465739237, −8.561095470453640663307113606317, −7.41840386094381557428359631295, −6.57151627396023210922579479916, −6.07008632277503567832840651619, −5.35049596646339340638296691584, −4.68404508156184007866714252843, −3.47316049738836127794905085806, −1.97791570942708712065464181645, −0.72822741090773578514497926136, 0.72822741090773578514497926136, 1.97791570942708712065464181645, 3.47316049738836127794905085806, 4.68404508156184007866714252843, 5.35049596646339340638296691584, 6.07008632277503567832840651619, 6.57151627396023210922579479916, 7.41840386094381557428359631295, 8.561095470453640663307113606317, 9.571906689743621188286465739237

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