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Properties

Label 1-1792-1792.109-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: $\chi_{1792} (109, \cdot )$
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

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

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