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Properties

Label 1-1792-1792.509-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.624 + 0.780i$
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  + (0.986 − 0.162i)3-s + (−0.227 − 0.973i)5-s + (0.946 − 0.321i)9-s + (0.412 − 0.910i)11-s + (0.290 + 0.956i)13-s + (−0.382 − 0.923i)15-s + (0.608 + 0.793i)17-s + (−0.0327 − 0.999i)19-s + (−0.442 + 0.896i)23-s + (−0.896 + 0.442i)25-s + (0.881 − 0.471i)27-s + (−0.0980 + 0.995i)29-s + (0.258 + 0.965i)31-s + (0.258 − 0.965i)33-s + (−0.528 + 0.849i)37-s + ⋯
L(s)  = 1  + (0.986 − 0.162i)3-s + (−0.227 − 0.973i)5-s + (0.946 − 0.321i)9-s + (0.412 − 0.910i)11-s + (0.290 + 0.956i)13-s + (−0.382 − 0.923i)15-s + (0.608 + 0.793i)17-s + (−0.0327 − 0.999i)19-s + (−0.442 + 0.896i)23-s + (−0.896 + 0.442i)25-s + (0.881 − 0.471i)27-s + (−0.0980 + 0.995i)29-s + (0.258 + 0.965i)31-s + (0.258 − 0.965i)33-s + (−0.528 + 0.849i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.624 + 0.780i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.624 + 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.437367875 + 1.171231649i\)
\(L(\frac12)\) \(\approx\) \(2.437367875 + 1.171231649i\)
\(L(1)\) \(\approx\) \(1.456517117 - 0.09739751744i\)
\(L(1)\) \(\approx\) \(1.456517117 - 0.09739751744i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.986 - 0.162i)T \)
5 \( 1 + (-0.227 - 0.973i)T \)
11 \( 1 + (0.412 - 0.910i)T \)
13 \( 1 + (0.290 + 0.956i)T \)
17 \( 1 + (0.608 + 0.793i)T \)
19 \( 1 + (-0.0327 - 0.999i)T \)
23 \( 1 + (-0.442 + 0.896i)T \)
29 \( 1 + (-0.0980 + 0.995i)T \)
31 \( 1 + (0.258 + 0.965i)T \)
37 \( 1 + (-0.528 + 0.849i)T \)
41 \( 1 + (-0.831 + 0.555i)T \)
43 \( 1 + (-0.634 + 0.773i)T \)
47 \( 1 + (0.130 + 0.991i)T \)
53 \( 1 + (0.910 + 0.412i)T \)
59 \( 1 + (0.683 + 0.729i)T \)
61 \( 1 + (-0.935 - 0.352i)T \)
67 \( 1 + (0.162 + 0.986i)T \)
71 \( 1 + (0.195 + 0.980i)T \)
73 \( 1 + (-0.659 + 0.751i)T \)
79 \( 1 + (0.793 + 0.608i)T \)
83 \( 1 + (0.471 - 0.881i)T \)
89 \( 1 + (-0.997 - 0.0654i)T \)
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

−19.98717830645401209097251461150, −19.16957515207744947479413801993, −18.49430050188382770523168611728, −18.03002009478381310295140966115, −16.93368574317664043892093087893, −16.03584286946059842558522165708, −15.10104473148111125037309785662, −14.969111295911805268157887510105, −14.02312542271328317528806551137, −13.50249437060622581029828126994, −12.37806433720413258733310421921, −11.842904933137702720419098158423, −10.5659591676198573856984019354, −10.14535351315690277455779019819, −9.46936343950578204786909440651, −8.36311448972206423513949078466, −7.7435335845195623598859848196, −7.107409507737104291845218506684, −6.212234873328851100050881968615, −5.12275892587302359458376020930, −3.95760373201946202616032625202, −3.53263395024336832933918768599, −2.50259438220974349537042956781, −1.8792597985189036139131622966, −0.40200036858900167139224857502, 1.21184760372892930304360541374, 1.50174285010112100133021005692, 2.913837980231829002624101337408, 3.68136078683617251162777692176, 4.40209298353430151709535081349, 5.36039577813458851262323657645, 6.42818904479438034665607540374, 7.22559513034987522033246141383, 8.26708063915097164109069773557, 8.65451051471603560807779485620, 9.29893302424192328672638692600, 10.1329460222978916209219015329, 11.293150215349722660849385544214, 11.98590333953335083904412116520, 12.806366099264228578577378472908, 13.54539437317484422802050306877, 14.03005792835627752948865194154, 14.919366618495784671171960125932, 15.746109543578143071391453772202, 16.35589490189911100440742808776, 17.05803181476749050742019451867, 18.012314957095119836257480909590, 18.94728600997599744143666871392, 19.49047229097095226462311943666, 19.96592133575141759056272996518

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