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Properties

Label 1-1792-1792.1285-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.987 + 0.156i$
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  + (0.227 + 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

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

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.987 + 0.156i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1285, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.987 + 0.156i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.385816509 + 0.1093717822i\)
\(L(\frac12)\) \(\approx\) \(1.385816509 + 0.1093717822i\)
\(L(1)\) \(\approx\) \(1.007713817 + 0.2827593003i\)
\(L(1)\) \(\approx\) \(1.007713817 + 0.2827593003i\)

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

−20.036647725878562998727967247825, −19.36627020065270043610508938367, −18.96429982625156567597973825054, −18.021142896441447309647226598672, −17.23337207743252511203000738695, −16.53154123939947697038703761209, −15.98557445456791365890396874292, −14.757143281683541270244472693357, −14.173942052557656172843639432900, −13.59166104576721803064887906227, −12.501247745921389438471614753874, −11.94356064346243216789971892349, −11.7653167442506320471165183806, −10.417183652889890102410832471401, −9.257909112021082540660381750804, −8.83950998617865828527845813295, −7.930567305874935579488908574776, −7.3293898470429693835841421658, −6.43219245035617768458124672935, −5.69151415484140224983719922444, −4.53510474333261174075469802017, −3.821445659636075339488222481374, −2.8028583670910603008765889427, −1.555744019257215899777145378957, −1.09304976387296917489412610517, 0.55451079752327535175770140509, 2.28731906982766015978797736915, 3.02034405684985603149125545744, 3.9405243413372459987986050130, 4.33699829371023755271696902344, 5.64576956448905457719129033622, 6.25356299666109535605999576979, 7.38769509886473700484110961686, 8.10432759047312693292758824596, 8.87112917591949692910447308862, 9.83089446056446280632656808538, 10.421314025784787604826698848891, 11.10561601252297334698068346479, 11.8374232326964432768934082376, 12.65782516933393401770629053035, 13.89445193251848096837223461465, 14.44597205608497772801062282644, 15.074226911833992922869355524, 15.679244142704049591434539937816, 16.368931453278855306951987208883, 17.36726451199667886949239904755, 17.78864347825430881453515763970, 19.12960956517356487268592541877, 19.4368477806852318995563437263, 20.1759971173745390131171608593

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