Properties

Label 2-1792-1.1-c1-0-42
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·3-s − 1.41·5-s + 7-s − 0.999·9-s − 2.82·11-s + 4.24·13-s − 2.00·15-s − 6·17-s − 4.24·19-s + 1.41·21-s − 6·23-s − 2.99·25-s − 5.65·27-s − 2.82·29-s + 4·31-s − 4.00·33-s − 1.41·35-s + 8.48·37-s + 6·39-s − 6·41-s + 8.48·43-s + 1.41·45-s + 49-s − 8.48·51-s − 5.65·53-s + 4.00·55-s − 6·57-s + ⋯
L(s)  = 1  + 0.816·3-s − 0.632·5-s + 0.377·7-s − 0.333·9-s − 0.852·11-s + 1.17·13-s − 0.516·15-s − 1.45·17-s − 0.973·19-s + 0.308·21-s − 1.25·23-s − 0.599·25-s − 1.08·27-s − 0.525·29-s + 0.718·31-s − 0.696·33-s − 0.239·35-s + 1.39·37-s + 0.960·39-s − 0.937·41-s + 1.29·43-s + 0.210·45-s + 0.142·49-s − 1.18·51-s − 0.777·53-s + 0.539·55-s − 0.794·57-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: \(1\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.41T + 3T^{2} \)
5 \( 1 + 1.41T + 5T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 8.48T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 8.48T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 5.65T + 53T^{2} \)
59 \( 1 - 1.41T + 59T^{2} \)
61 \( 1 + 12.7T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−8.694475224141748518501503378748, −8.109684716507400585064540236901, −7.71216734626173517792982431606, −6.44583289603840765441198347727, −5.76427495609112689575169259534, −4.44630372820368743640760102218, −3.90322769668473811996054539871, −2.77556981339043928521130820304, −1.92428702002095385385325786911, 0, 1.92428702002095385385325786911, 2.77556981339043928521130820304, 3.90322769668473811996054539871, 4.44630372820368743640760102218, 5.76427495609112689575169259534, 6.44583289603840765441198347727, 7.71216734626173517792982431606, 8.109684716507400585064540236901, 8.694475224141748518501503378748

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