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Properties

Label 1-1792-1792.1381-r0-0-0
Degree $1$
Conductor $1792$
Sign $-0.480 - 0.877i$
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  + (−0.0327 + 0.999i)3-s + (0.352 − 0.935i)5-s + (−0.997 − 0.0654i)9-s + (−0.227 + 0.973i)11-s + (−0.773 − 0.634i)13-s + (0.923 + 0.382i)15-s + (−0.130 + 0.991i)17-s + (0.812 − 0.582i)19-s + (−0.659 − 0.751i)23-s + (−0.751 − 0.659i)25-s + (0.0980 − 0.995i)27-s + (−0.290 + 0.956i)29-s + (−0.965 + 0.258i)31-s + (−0.965 − 0.258i)33-s + (0.412 + 0.910i)37-s + ⋯
L(s)  = 1  + (−0.0327 + 0.999i)3-s + (0.352 − 0.935i)5-s + (−0.997 − 0.0654i)9-s + (−0.227 + 0.973i)11-s + (−0.773 − 0.634i)13-s + (0.923 + 0.382i)15-s + (−0.130 + 0.991i)17-s + (0.812 − 0.582i)19-s + (−0.659 − 0.751i)23-s + (−0.751 − 0.659i)25-s + (0.0980 − 0.995i)27-s + (−0.290 + 0.956i)29-s + (−0.965 + 0.258i)31-s + (−0.965 − 0.258i)33-s + (0.412 + 0.910i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.480 - 0.877i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ -0.480 - 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1905626987 - 0.3216638601i\)
\(L(\frac12)\) \(\approx\) \(0.1905626987 - 0.3216638601i\)
\(L(1)\) \(\approx\) \(0.8235000488 + 0.09772618557i\)
\(L(1)\) \(\approx\) \(0.8235000488 + 0.09772618557i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.0327 + 0.999i)T \)
5 \( 1 + (0.352 - 0.935i)T \)
11 \( 1 + (-0.227 + 0.973i)T \)
13 \( 1 + (-0.773 - 0.634i)T \)
17 \( 1 + (-0.130 + 0.991i)T \)
19 \( 1 + (0.812 - 0.582i)T \)
23 \( 1 + (-0.659 - 0.751i)T \)
29 \( 1 + (-0.290 + 0.956i)T \)
31 \( 1 + (-0.965 + 0.258i)T \)
37 \( 1 + (0.412 + 0.910i)T \)
41 \( 1 + (-0.195 - 0.980i)T \)
43 \( 1 + (-0.881 - 0.471i)T \)
47 \( 1 + (0.608 - 0.793i)T \)
53 \( 1 + (0.973 + 0.227i)T \)
59 \( 1 + (-0.162 - 0.986i)T \)
61 \( 1 + (-0.528 + 0.849i)T \)
67 \( 1 + (-0.999 - 0.0327i)T \)
71 \( 1 + (-0.555 - 0.831i)T \)
73 \( 1 + (0.896 - 0.442i)T \)
79 \( 1 + (0.991 - 0.130i)T \)
83 \( 1 + (-0.995 + 0.0980i)T \)
89 \( 1 + (-0.321 - 0.946i)T \)
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

−20.25846852500016814227573930539, −19.526621376216098397911758025502, −18.85824802570464878088858633692, −18.27617803102239624347005409982, −17.81110976986926896854662216470, −16.79965278030590586594353960561, −16.21832443191606974880735737971, −15.062858460929863077753706424240, −14.29276747153994837801926428702, −13.77452337561160160994600121723, −13.25229575485712318348901079900, −12.08178523046684066935784323227, −11.53034194291330864415275938395, −10.940721262043920490176661348841, −9.78036712102776900889470595776, −9.22691581227027675305209487392, −7.95880830808599758139344090953, −7.48802454446437153673248227985, −6.73158486961359918519882950900, −5.88783136162768987771543296024, −5.35466527718710093888357152519, −3.88233322428470230391619651090, −2.908660270026299644176296225324, −2.3197878381828918757718402472, −1.30629966125608347907155700515, 0.12631314372783956267358693010, 1.64374240183287901575584339546, 2.58608308928750183787681111139, 3.676666808402218306052424995559, 4.544531544673607230576350188522, 5.15994691942104852746254927145, 5.74585826759863444116848058909, 6.956942029309130741490961245054, 7.96215407988098372877322228128, 8.75992848713913312399740617647, 9.41484473126277128678985108541, 10.192469650372181358667251427064, 10.62445622813959236488804515123, 11.90258913220667046739753718589, 12.40407312232203640641599259198, 13.2261994075132171085604501134, 14.1192166557872910723395576250, 15.05684629968069975460397578705, 15.38978863912103276696073698803, 16.44210078600436841693520216534, 16.85099338399267148277343406659, 17.65051182607773900368429187140, 18.21080420220674626493985827896, 19.6559059539075849756951784560, 20.21175738718764304162618156802

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