Properties

Label 1-1792-1792.123-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.156 + 0.987i$
Analytic cond. $192.577$
Root an. cond. $192.577$
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  + (0.227 + 0.973i)3-s + (0.582 − 0.812i)5-s + (−0.896 + 0.442i)9-s + (0.999 − 0.0327i)11-s + (0.0980 + 0.995i)13-s + (0.923 + 0.382i)15-s + (0.793 + 0.608i)17-s + (−0.352 − 0.935i)19-s + (0.946 + 0.321i)23-s + (−0.321 − 0.946i)25-s + (−0.634 − 0.773i)27-s + (−0.881 + 0.471i)29-s + (−0.258 + 0.965i)31-s + (0.258 + 0.965i)33-s + (0.162 + 0.986i)37-s + ⋯
L(s)  = 1  + (0.227 + 0.973i)3-s + (0.582 − 0.812i)5-s + (−0.896 + 0.442i)9-s + (0.999 − 0.0327i)11-s + (0.0980 + 0.995i)13-s + (0.923 + 0.382i)15-s + (0.793 + 0.608i)17-s + (−0.352 − 0.935i)19-s + (0.946 + 0.321i)23-s + (−0.321 − 0.946i)25-s + (−0.634 − 0.773i)27-s + (−0.881 + 0.471i)29-s + (−0.258 + 0.965i)31-s + (0.258 + 0.965i)33-s + (0.162 + 0.986i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.156 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.808131412 + 2.117989804i\)
\(L(\frac12)\) \(\approx\) \(1.808131412 + 2.117989804i\)
\(L(1)\) \(\approx\) \(1.285500702 + 0.4413203858i\)
\(L(1)\) \(\approx\) \(1.285500702 + 0.4413203858i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.227 + 0.973i)T \)
5 \( 1 + (0.582 - 0.812i)T \)
11 \( 1 + (0.999 - 0.0327i)T \)
13 \( 1 + (0.0980 + 0.995i)T \)
17 \( 1 + (0.793 + 0.608i)T \)
19 \( 1 + (-0.352 - 0.935i)T \)
23 \( 1 + (0.946 + 0.321i)T \)
29 \( 1 + (-0.881 + 0.471i)T \)
31 \( 1 + (-0.258 + 0.965i)T \)
37 \( 1 + (0.162 + 0.986i)T \)
41 \( 1 + (0.980 - 0.195i)T \)
43 \( 1 + (0.956 - 0.290i)T \)
47 \( 1 + (-0.991 - 0.130i)T \)
53 \( 1 + (0.0327 + 0.999i)T \)
59 \( 1 + (0.910 - 0.412i)T \)
61 \( 1 + (0.683 + 0.729i)T \)
67 \( 1 + (-0.973 + 0.227i)T \)
71 \( 1 + (0.831 - 0.555i)T \)
73 \( 1 + (-0.997 - 0.0654i)T \)
79 \( 1 + (-0.608 - 0.793i)T \)
83 \( 1 + (-0.773 - 0.634i)T \)
89 \( 1 + (0.751 + 0.659i)T \)
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

−19.57920588190107851737851951500, −19.02520080494831716020471402050, −18.4113411241454322419304634220, −17.69853074255215147654674191399, −17.12363070223480525428354204734, −16.307525708370730586559707013854, −14.89129973542410556660203523211, −14.65620723224559949973481391183, −13.96499223113961123621698403603, −13.03029192115829600664473253081, −12.58690493263229635692152399839, −11.51139328016840673531006270805, −11.003561846555359869203739949365, −9.87597290251824464463265592351, −9.30805636643552216321116352963, −8.24863187645811985267224412014, −7.49327873227274572777674386666, −6.86893263959759130612085653209, −5.934754636360166780886314315863, −5.567440572437937579310178465851, −3.95640412148861012680918557372, −3.13159442380856555003650988946, −2.359999244493930857377151945059, −1.44497145515035103179933670415, −0.514478168383688850549633270089, 1.02328473591134281646018737292, 1.88174839536052520390743769447, 3.03899718416217876516947566147, 3.98637083313083072469595727136, 4.62652688138313147950472854794, 5.42956049296507401273637287650, 6.23035982843344595637854622049, 7.211667444148974147882054112586, 8.45645918557104515211160134529, 9.06648103378958417413342347937, 9.40888960887577878026592440272, 10.3521942284071302700955647626, 11.186457108834152713190995835482, 11.91059104865020930558307978577, 12.85334244320368423364776189222, 13.64397908646472809600200429158, 14.43043608645864274038582513436, 14.91007260989184150820799642801, 15.985931639968828332219226852182, 16.563254766136823500991360062012, 17.10941555497104808765590636898, 17.69531719688333146857022287893, 19.06852423630321022426514320172, 19.52246272560949770605309413535, 20.37833700725338396506119858237

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