Properties

Label 2-1792-1792.1413-c0-0-0
Degree $2$
Conductor $1792$
Sign $0.870 - 0.492i$
Analytic cond. $0.894324$
Root an. cond. $0.945687$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.773 + 0.634i)2-s + (0.195 − 0.980i)4-s + (0.881 + 0.471i)7-s + (0.471 + 0.881i)8-s + (0.956 + 0.290i)9-s + (1.55 − 0.929i)11-s + (−0.980 + 0.195i)14-s + (−0.923 − 0.382i)16-s + (−0.923 + 0.382i)18-s + (−0.609 + 1.70i)22-s + (−0.448 − 0.368i)23-s + (−0.995 − 0.0980i)25-s + (0.634 − 0.773i)28-s + (−0.485 − 1.93i)29-s + (0.956 − 0.290i)32-s + ⋯
L(s)  = 1  + (−0.773 + 0.634i)2-s + (0.195 − 0.980i)4-s + (0.881 + 0.471i)7-s + (0.471 + 0.881i)8-s + (0.956 + 0.290i)9-s + (1.55 − 0.929i)11-s + (−0.980 + 0.195i)14-s + (−0.923 − 0.382i)16-s + (−0.923 + 0.382i)18-s + (−0.609 + 1.70i)22-s + (−0.448 − 0.368i)23-s + (−0.995 − 0.0980i)25-s + (0.634 − 0.773i)28-s + (−0.485 − 1.93i)29-s + (0.956 − 0.290i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.870 - 0.492i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ 0.870 - 0.492i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9942798980\)
\(L(\frac12)\) \(\approx\) \(0.9942798980\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.773 - 0.634i)T \)
7 \( 1 + (-0.881 - 0.471i)T \)
good3 \( 1 + (-0.956 - 0.290i)T^{2} \)
5 \( 1 + (0.995 + 0.0980i)T^{2} \)
11 \( 1 + (-1.55 + 0.929i)T + (0.471 - 0.881i)T^{2} \)
13 \( 1 + (-0.0980 - 0.995i)T^{2} \)
17 \( 1 + (0.923 - 0.382i)T^{2} \)
19 \( 1 + (-0.634 + 0.773i)T^{2} \)
23 \( 1 + (0.448 + 0.368i)T + (0.195 + 0.980i)T^{2} \)
29 \( 1 + (0.485 + 1.93i)T + (-0.881 + 0.471i)T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (1.61 - 0.577i)T + (0.773 - 0.634i)T^{2} \)
41 \( 1 + (-0.980 + 0.195i)T^{2} \)
43 \( 1 + (-0.284 - 1.91i)T + (-0.956 + 0.290i)T^{2} \)
47 \( 1 + (0.382 + 0.923i)T^{2} \)
53 \( 1 + (-0.289 + 1.15i)T + (-0.881 - 0.471i)T^{2} \)
59 \( 1 + (-0.0980 + 0.995i)T^{2} \)
61 \( 1 + (-0.290 + 0.956i)T^{2} \)
67 \( 1 + (0.174 + 0.235i)T + (-0.290 + 0.956i)T^{2} \)
71 \( 1 + (-0.482 - 1.59i)T + (-0.831 + 0.555i)T^{2} \)
73 \( 1 + (-0.555 + 0.831i)T^{2} \)
79 \( 1 + (-0.162 + 0.108i)T + (0.382 - 0.923i)T^{2} \)
83 \( 1 + (-0.773 - 0.634i)T^{2} \)
89 \( 1 + (-0.195 + 0.980i)T^{2} \)
97 \( 1 + (0.707 + 0.707i)T^{2} \)
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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

−9.539701366314971023218822285767, −8.547418768418727022370730915945, −8.131824273895228791941254137463, −7.26469828120423148770558087720, −6.36955206604365089720224015367, −5.79111218302425120706752004621, −4.71336178450722289812414315735, −3.89738040399808205854146903833, −2.15212037076379916704565136263, −1.26799675683809781506356046889, 1.41040070711846495948599374404, 1.89579926886443591809887114242, 3.73808837823134575534172376315, 4.01874654420244531801499708742, 5.14137654125868567953100891437, 6.66169031054577801291504252553, 7.20194542036143539428746689405, 7.77716979448488955133045330442, 9.022406903763176216666303858101, 9.211222990820565049426595828826

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