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Properties

Label 1-1792-1792.773-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.643 + 0.765i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.643 + 0.765i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (773, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.643 + 0.765i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9271557140 + 0.4315374068i\)
\(L(\frac12)\) \(\approx\) \(0.9271557140 + 0.4315374068i\)
\(L(1)\) \(\approx\) \(0.9310906732 - 0.3205775847i\)
\(L(1)\) \(\approx\) \(0.9310906732 - 0.3205775847i\)

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

−20.001314521835518473172282216437, −19.36419494147120949402586838891, −18.1861841349931656064143181466, −17.584244404647314552799776121867, −17.16974796058505467888648560648, −16.16910655524960893765300452458, −15.28355211916151962840800061330, −14.99703213375199245802353556449, −14.03002418713625940045862878060, −13.47294894269795701369531029540, −12.31935244118280218869784981638, −11.43295629019161212838455418319, −10.85114986946385843559060526530, −10.14442765066818376395252949960, −9.48569566085260574026834008496, −8.76082303302339690954265333080, −7.756441268926415004017270467808, −6.55055064368112639704458110705, −6.15756923508277475293398646865, −5.20942142652764880514588816219, −4.318246810848632189267046147376, −3.42561285167924541832593568095, −2.751138051505700607559292689437, −1.58122711806311802319032379429, −0.1970687669710816907597287356, 0.97287026482336262153344042161, 1.69445901721425197182212745294, 2.39377440622083622752346051554, 3.79176538831463905421719710097, 4.68957124354833588719019043434, 5.59736485282252483544743438258, 6.40614426122452124636763363993, 6.89546198277321769337814996179, 8.018299938673398637351773571814, 8.75324025843846530033221269609, 9.35133633697956920058751841871, 10.31674874592553884417118181655, 11.4478758649010150078633860601, 12.03135884213317422188566643701, 12.52520068867752752889691086311, 13.59364718902362055964785601345, 13.9448044276905455824834131005, 14.610176914673621196810658693050, 16.20562081824833094572762820232, 16.34089658421460568086362481089, 17.35796021426118931224412131560, 17.83245768438400059370241654235, 18.55801610518491967827745690506, 19.51008487397858627882280737555, 19.92649316680813034457722073722

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