Properties

Label 2-1792-1792.181-c0-0-0
Degree $2$
Conductor $1792$
Sign $0.997 + 0.0735i$
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  + (0.956 − 0.290i)2-s + (0.831 − 0.555i)4-s + (−0.773 + 0.634i)7-s + (0.634 − 0.773i)8-s + (0.0980 + 0.995i)9-s + (1.07 + 0.509i)11-s + (−0.555 + 0.831i)14-s + (0.382 − 0.923i)16-s + (0.382 + 0.923i)18-s + (1.17 + 0.174i)22-s + (1.90 + 0.577i)23-s + (−0.881 − 0.471i)25-s + (−0.290 + 0.956i)28-s + (−1.82 − 0.653i)29-s + (0.0980 − 0.995i)32-s + ⋯
L(s)  = 1  + (0.956 − 0.290i)2-s + (0.831 − 0.555i)4-s + (−0.773 + 0.634i)7-s + (0.634 − 0.773i)8-s + (0.0980 + 0.995i)9-s + (1.07 + 0.509i)11-s + (−0.555 + 0.831i)14-s + (0.382 − 0.923i)16-s + (0.382 + 0.923i)18-s + (1.17 + 0.174i)22-s + (1.90 + 0.577i)23-s + (−0.881 − 0.471i)25-s + (−0.290 + 0.956i)28-s + (−1.82 − 0.653i)29-s + (0.0980 − 0.995i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.997 + 0.0735i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ 0.997 + 0.0735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.030398685\)
\(L(\frac12)\) \(\approx\) \(2.030398685\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.956 + 0.290i)T \)
7 \( 1 + (0.773 - 0.634i)T \)
good3 \( 1 + (-0.0980 - 0.995i)T^{2} \)
5 \( 1 + (0.881 + 0.471i)T^{2} \)
11 \( 1 + (-1.07 - 0.509i)T + (0.634 + 0.773i)T^{2} \)
13 \( 1 + (-0.471 - 0.881i)T^{2} \)
17 \( 1 + (-0.382 - 0.923i)T^{2} \)
19 \( 1 + (0.290 - 0.956i)T^{2} \)
23 \( 1 + (-1.90 - 0.577i)T + (0.831 + 0.555i)T^{2} \)
29 \( 1 + (1.82 + 0.653i)T + (0.773 + 0.634i)T^{2} \)
31 \( 1 + (0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.265 + 1.78i)T + (-0.956 + 0.290i)T^{2} \)
41 \( 1 + (-0.555 + 0.831i)T^{2} \)
43 \( 1 + (-0.452 - 0.499i)T + (-0.0980 + 0.995i)T^{2} \)
47 \( 1 + (0.923 - 0.382i)T^{2} \)
53 \( 1 + (0.0923 - 0.0330i)T + (0.773 - 0.634i)T^{2} \)
59 \( 1 + (-0.471 + 0.881i)T^{2} \)
61 \( 1 + (-0.995 + 0.0980i)T^{2} \)
67 \( 1 + (-0.0659 - 1.34i)T + (-0.995 + 0.0980i)T^{2} \)
71 \( 1 + (1.95 + 0.192i)T + (0.980 + 0.195i)T^{2} \)
73 \( 1 + (-0.195 - 0.980i)T^{2} \)
79 \( 1 + (0.924 + 0.183i)T + (0.923 + 0.382i)T^{2} \)
83 \( 1 + (0.956 + 0.290i)T^{2} \)
89 \( 1 + (-0.831 + 0.555i)T^{2} \)
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

−9.517238689586608653952014630604, −8.926463706702192034631386801332, −7.50147457700071518319965363871, −7.07087404647991792999342378880, −5.98573327322812623840734781135, −5.47565230899576142080643900383, −4.43591643531319919248561969373, −3.64967220949187080329605482082, −2.59133477521843441709169141117, −1.69520971961813026493590070480, 1.36852538111474267944549193473, 3.10030257964995933707039328605, 3.58030784443890810909283118043, 4.39622561375101540860269177971, 5.52563045183091122082311434505, 6.37261984646165457223771412679, 6.86080023842456609235385377897, 7.54060680324377187256624392381, 8.815729263537130188629326889107, 9.317929649443466612809045485220

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