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Properties

Label 1-1792-1792.1045-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.996 + 0.0878i$
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  + (0.935 − 0.352i)3-s + (−0.683 + 0.729i)5-s + (0.751 − 0.659i)9-s + (0.812 + 0.582i)11-s + (0.956 + 0.290i)13-s + (−0.382 + 0.923i)15-s + (0.991 + 0.130i)17-s + (0.528 − 0.849i)19-s + (−0.997 − 0.0654i)23-s + (−0.0654 − 0.997i)25-s + (0.471 − 0.881i)27-s + (−0.995 + 0.0980i)29-s + (0.965 − 0.258i)31-s + (0.965 + 0.258i)33-s + (0.0327 + 0.999i)37-s + ⋯
L(s)  = 1  + (0.935 − 0.352i)3-s + (−0.683 + 0.729i)5-s + (0.751 − 0.659i)9-s + (0.812 + 0.582i)11-s + (0.956 + 0.290i)13-s + (−0.382 + 0.923i)15-s + (0.991 + 0.130i)17-s + (0.528 − 0.849i)19-s + (−0.997 − 0.0654i)23-s + (−0.0654 − 0.997i)25-s + (0.471 − 0.881i)27-s + (−0.995 + 0.0980i)29-s + (0.965 − 0.258i)31-s + (0.965 + 0.258i)33-s + (0.0327 + 0.999i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.996 + 0.0878i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1045, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.996 + 0.0878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.377758011 + 0.1045923081i\)
\(L(\frac12)\) \(\approx\) \(2.377758011 + 0.1045923081i\)
\(L(1)\) \(\approx\) \(1.499449011 + 0.01645857547i\)
\(L(1)\) \(\approx\) \(1.499449011 + 0.01645857547i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.935 - 0.352i)T \)
5 \( 1 + (-0.683 + 0.729i)T \)
11 \( 1 + (0.812 + 0.582i)T \)
13 \( 1 + (0.956 + 0.290i)T \)
17 \( 1 + (0.991 + 0.130i)T \)
19 \( 1 + (0.528 - 0.849i)T \)
23 \( 1 + (-0.997 - 0.0654i)T \)
29 \( 1 + (-0.995 + 0.0980i)T \)
31 \( 1 + (0.965 - 0.258i)T \)
37 \( 1 + (0.0327 + 0.999i)T \)
41 \( 1 + (-0.831 - 0.555i)T \)
43 \( 1 + (0.773 - 0.634i)T \)
47 \( 1 + (-0.793 - 0.608i)T \)
53 \( 1 + (0.582 - 0.812i)T \)
59 \( 1 + (-0.227 + 0.973i)T \)
61 \( 1 + (0.986 + 0.162i)T \)
67 \( 1 + (0.352 + 0.935i)T \)
71 \( 1 + (-0.195 + 0.980i)T \)
73 \( 1 + (-0.321 + 0.946i)T \)
79 \( 1 + (0.130 + 0.991i)T \)
83 \( 1 + (-0.881 + 0.471i)T \)
89 \( 1 + (0.442 - 0.896i)T \)
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

−20.25322749792297334652177621025, −19.52552478200477588178631845090, −18.942835360706328280763004053287, −18.209481134704262454803195306721, −17.0098063681661582284524441693, −16.18960489516246451612542124484, −16.00098476802133415229102185516, −14.98478911357225346538758285941, −14.25860603422216430910205612125, −13.652430433166153941652265181505, −12.79084766330178891777449000990, −12.00977296601613717052742669436, −11.26657629984519234082696978393, −10.278977088794151861400515752658, −9.4426568205344638800562695711, −8.83404595362312523078758218058, −7.93206334005136867898899808974, −7.73832093665386744929973393689, −6.31238312445842888807458732266, −5.45816231141065331451569111425, −4.42535972361276849035108526373, −3.61942466204768192317463782193, −3.28372858403276435337765857094, −1.77278818269968662797088629912, −0.99510170615167932953618400199, 1.02172794042716246610182552000, 1.999869508385597562843862711491, 2.97894614394439880170428531854, 3.761297546687300258815755487842, 4.25806308442351023594255366736, 5.71209647312721438773402592348, 6.79764438048188908787993792, 7.12015238387571789664890972476, 8.12181347178996781718424888030, 8.64431719694587825284337680585, 9.69967797437498399407591911261, 10.22880202686806100193631821749, 11.52058152608207341787815774030, 11.84268855653725949450594871863, 12.83663942476487217601173051095, 13.742337953460632726577549742190, 14.25885047654800555804681388082, 15.020971728707904470242186508819, 15.55767117741512019392903714484, 16.36683124225054917117029593465, 17.45320347857781314445355869975, 18.27104059916474394066055628166, 18.8271484893532556485940817457, 19.42154932660278056749123454997, 20.195633281730917762644190339627

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