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Properties

Label 1-1792-1792.299-r0-0-0
Degree $1$
Conductor $1792$
Sign $-0.810 + 0.585i$
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  + (−0.352 − 0.935i)3-s + (0.729 + 0.683i)5-s + (−0.751 + 0.659i)9-s + (0.582 − 0.812i)11-s + (−0.290 + 0.956i)13-s + (0.382 − 0.923i)15-s + (−0.991 − 0.130i)17-s + (−0.849 − 0.528i)19-s + (−0.997 − 0.0654i)23-s + (0.0654 + 0.997i)25-s + (0.881 + 0.471i)27-s + (0.0980 + 0.995i)29-s + (0.965 − 0.258i)31-s + (−0.965 − 0.258i)33-s + (−0.999 + 0.0327i)37-s + ⋯
L(s)  = 1  + (−0.352 − 0.935i)3-s + (0.729 + 0.683i)5-s + (−0.751 + 0.659i)9-s + (0.582 − 0.812i)11-s + (−0.290 + 0.956i)13-s + (0.382 − 0.923i)15-s + (−0.991 − 0.130i)17-s + (−0.849 − 0.528i)19-s + (−0.997 − 0.0654i)23-s + (0.0654 + 0.997i)25-s + (0.881 + 0.471i)27-s + (0.0980 + 0.995i)29-s + (0.965 − 0.258i)31-s + (−0.965 − 0.258i)33-s + (−0.999 + 0.0327i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.810 + 0.585i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ -0.810 + 0.585i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04321269101 + 0.1335658087i\)
\(L(\frac12)\) \(\approx\) \(0.04321269101 + 0.1335658087i\)
\(L(1)\) \(\approx\) \(0.7948549536 - 0.1065417405i\)
\(L(1)\) \(\approx\) \(0.7948549536 - 0.1065417405i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.352 - 0.935i)T \)
5 \( 1 + (0.729 + 0.683i)T \)
11 \( 1 + (0.582 - 0.812i)T \)
13 \( 1 + (-0.290 + 0.956i)T \)
17 \( 1 + (-0.991 - 0.130i)T \)
19 \( 1 + (-0.849 - 0.528i)T \)
23 \( 1 + (-0.997 - 0.0654i)T \)
29 \( 1 + (0.0980 + 0.995i)T \)
31 \( 1 + (0.965 - 0.258i)T \)
37 \( 1 + (-0.999 + 0.0327i)T \)
41 \( 1 + (-0.831 - 0.555i)T \)
43 \( 1 + (-0.634 - 0.773i)T \)
47 \( 1 + (-0.793 - 0.608i)T \)
53 \( 1 + (0.812 + 0.582i)T \)
59 \( 1 + (-0.973 - 0.227i)T \)
61 \( 1 + (-0.162 + 0.986i)T \)
67 \( 1 + (-0.935 + 0.352i)T \)
71 \( 1 + (-0.195 + 0.980i)T \)
73 \( 1 + (-0.321 + 0.946i)T \)
79 \( 1 + (-0.130 - 0.991i)T \)
83 \( 1 + (0.471 + 0.881i)T \)
89 \( 1 + (0.442 - 0.896i)T \)
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

−20.04759161780796505555460343355, −19.49719368139931071417530218741, −17.9962426094764326784205213282, −17.60914636343236479191289518764, −17.02380645579255514221720028539, −16.31841894182649559137216917263, −15.40868093028833126820827015516, −14.96119973628548426219574217530, −14.01434597619096473229542945684, −13.202489865057680581258173091460, −12.31637871795166686759633171921, −11.802938782424108701961787584982, −10.63637148236747821116047145627, −10.063011567052111353012751371065, −9.52328056646530260693226864599, −8.64588950421048108526537866862, −7.96812419644493297156513963478, −6.449082037566642016989160964522, −6.11869772528677364162153631369, −4.95656922065902088643338111033, −4.5750055539601490104900229985, −3.63937737690637404086172172766, −2.45341914931435325985520361259, −1.52846210807473681025763122620, −0.047556341615929853812679991467, 1.49420896550657940297972439300, 2.14080313979464023703354119302, 2.989010866553413179585035310003, 4.179527801135518040293646994923, 5.25673878852545928978555608853, 6.15594779128306837062354741569, 6.706564146509290319868702995387, 7.16488520529649172425126174764, 8.526740286836744570372158984990, 8.916605940403030142331167782313, 10.13101089230671319914638473881, 10.84369267590062076435485115489, 11.59611352448756709508088996507, 12.165749302784890535492055937018, 13.32335922802685081151077025372, 13.720031809221268228499163307, 14.314457608043190868550322546783, 15.2028043955186353484153248694, 16.31574313506560723032032302600, 17.03571686769379874620405105602, 17.57738262758310006020029752611, 18.317041562485973257228042310873, 18.9842500186632069752112291713, 19.49994640759698555855039401949, 20.35337044910900424931345688704

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