Properties

Label 1-1792-1792.1459-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.416 - 0.909i$
Analytic cond. $8.32201$
Root an. cond. $8.32201$
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  + (−0.729 − 0.683i)3-s + (−0.412 + 0.910i)5-s + (0.0654 + 0.997i)9-s + (−0.528 + 0.849i)11-s + (−0.0980 + 0.995i)13-s + (0.923 − 0.382i)15-s + (−0.130 − 0.991i)17-s + (−0.986 + 0.162i)19-s + (−0.751 − 0.659i)23-s + (−0.659 − 0.751i)25-s + (0.634 − 0.773i)27-s + (−0.881 − 0.471i)29-s + (−0.965 − 0.258i)31-s + (0.965 − 0.258i)33-s + (−0.935 + 0.352i)37-s + ⋯
L(s)  = 1  + (−0.729 − 0.683i)3-s + (−0.412 + 0.910i)5-s + (0.0654 + 0.997i)9-s + (−0.528 + 0.849i)11-s + (−0.0980 + 0.995i)13-s + (0.923 − 0.382i)15-s + (−0.130 − 0.991i)17-s + (−0.986 + 0.162i)19-s + (−0.751 − 0.659i)23-s + (−0.659 − 0.751i)25-s + (0.634 − 0.773i)27-s + (−0.881 − 0.471i)29-s + (−0.965 − 0.258i)31-s + (0.965 − 0.258i)33-s + (−0.935 + 0.352i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.416 - 0.909i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.416 - 0.909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4080021490 - 0.2619024919i\)
\(L(\frac12)\) \(\approx\) \(0.4080021490 - 0.2619024919i\)
\(L(1)\) \(\approx\) \(0.6213160314 + 0.004899936354i\)
\(L(1)\) \(\approx\) \(0.6213160314 + 0.004899936354i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.729 - 0.683i)T \)
5 \( 1 + (-0.412 + 0.910i)T \)
11 \( 1 + (-0.528 + 0.849i)T \)
13 \( 1 + (-0.0980 + 0.995i)T \)
17 \( 1 + (-0.130 - 0.991i)T \)
19 \( 1 + (-0.986 + 0.162i)T \)
23 \( 1 + (-0.751 - 0.659i)T \)
29 \( 1 + (-0.881 - 0.471i)T \)
31 \( 1 + (-0.965 - 0.258i)T \)
37 \( 1 + (-0.935 + 0.352i)T \)
41 \( 1 + (-0.980 - 0.195i)T \)
43 \( 1 + (0.956 + 0.290i)T \)
47 \( 1 + (-0.608 - 0.793i)T \)
53 \( 1 + (0.849 + 0.528i)T \)
59 \( 1 + (0.812 - 0.582i)T \)
61 \( 1 + (0.973 + 0.227i)T \)
67 \( 1 + (0.683 - 0.729i)T \)
71 \( 1 + (0.831 + 0.555i)T \)
73 \( 1 + (-0.442 + 0.896i)T \)
79 \( 1 + (0.991 + 0.130i)T \)
83 \( 1 + (0.773 - 0.634i)T \)
89 \( 1 + (0.946 + 0.321i)T \)
97 \( 1 + (0.707 - 0.707i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.46390214370278446847024186482, −19.68531434193324953533517482460, −18.95257353939635365788880679303, −17.832528634170978575342183059611, −17.373050451683367899355749704, −16.51929637922822569049450551246, −16.06528349765789339765502144888, −15.30418208566360529049533097825, −14.729519644800835785565039182909, −13.40909652692376626803533559303, −12.769959975234229218428573932583, −12.19049049895310020010793527797, −11.18554873687686554100758908784, −10.70279093225284237442203874117, −9.90889694512574203562890053160, −8.8932765360946059951170903308, −8.371176863864755023208115584681, −7.462058733391666516394322372929, −6.24755685721285689259906176538, −5.4973644708081025863212185877, −5.04984999418447746076347394527, −3.86430683719019605498611462947, −3.511945266203636332336282143737, −1.95167426658556145497226466950, −0.71025137340766597128493754916, 0.27472959424268957322974265779, 2.05688634423509319435931878977, 2.25654360405705630242946528298, 3.73277136781699520841928765631, 4.57442032710419306995382784727, 5.46621022813986505225899651909, 6.49997719605948764017113795454, 6.98432038418198921026098466837, 7.60654302739775476549642047751, 8.50142519279759496998770475239, 9.72218801213821328498112460341, 10.434959888344065601116838952953, 11.19958737959374222090127810419, 11.80820954033164142235848477895, 12.47747940876830262913611934568, 13.33210424105346962283804687906, 14.13361005504943042254743394813, 14.8446136005765230221560829442, 15.71715190981921006132930271494, 16.44760938025819081861329835606, 17.2206713408276262788235665458, 18.03642947947885078189887189217, 18.60706660502151289683775973042, 19.023952281802650732228133646811, 19.94778489552268465815586797716

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