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Properties

Label 1-1392-1392.1283-r0-0-0
Degree $1$
Conductor $1392$
Sign $0.743 - 0.669i$
Analytic cond. $6.46442$
Root an. cond. $6.46442$
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  + (−0.433 − 0.900i)5-s + (0.623 + 0.781i)7-s + (−0.974 + 0.222i)11-s + (0.974 − 0.222i)13-s − 17-s + (−0.781 − 0.623i)19-s + (0.900 + 0.433i)23-s + (−0.623 + 0.781i)25-s + (0.900 − 0.433i)31-s + (0.433 − 0.900i)35-s + (0.974 + 0.222i)37-s + 41-s + (0.433 − 0.900i)43-s + (−0.222 − 0.974i)47-s + (−0.222 + 0.974i)49-s + ⋯
L(s)  = 1  + (−0.433 − 0.900i)5-s + (0.623 + 0.781i)7-s + (−0.974 + 0.222i)11-s + (0.974 − 0.222i)13-s − 17-s + (−0.781 − 0.623i)19-s + (0.900 + 0.433i)23-s + (−0.623 + 0.781i)25-s + (0.900 − 0.433i)31-s + (0.433 − 0.900i)35-s + (0.974 + 0.222i)37-s + 41-s + (0.433 − 0.900i)43-s + (−0.222 − 0.974i)47-s + (−0.222 + 0.974i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1392 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1392 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1392\)    =    \(2^{4} \cdot 3 \cdot 29\)
Sign: $0.743 - 0.669i$
Analytic conductor: \(6.46442\)
Root analytic conductor: \(6.46442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1392} (1283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1392,\ (0:\ ),\ 0.743 - 0.669i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.276116058 - 0.4899579037i\)
\(L(\frac12)\) \(\approx\) \(1.276116058 - 0.4899579037i\)
\(L(1)\) \(\approx\) \(1.018404986 - 0.1323562790i\)
\(L(1)\) \(\approx\) \(1.018404986 - 0.1323562790i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
29 \( 1 \)
good5 \( 1 + (-0.433 - 0.900i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (-0.974 + 0.222i)T \)
13 \( 1 + (0.974 - 0.222i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.781 - 0.623i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (0.900 - 0.433i)T \)
37 \( 1 + (0.974 + 0.222i)T \)
41 \( 1 + T \)
43 \( 1 + (0.433 - 0.900i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (-0.433 - 0.900i)T \)
59 \( 1 - iT \)
61 \( 1 + (-0.781 + 0.623i)T \)
67 \( 1 + (0.974 + 0.222i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + (0.222 - 0.974i)T \)
83 \( 1 + (0.781 + 0.623i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−21.129417195967792496395259631035, −20.11659074157457614026321499912, −19.35446378058452819651468461497, −18.544849782349371812465683249992, −18.030763853442942200784120762927, −17.212584525605981918855489114612, −16.247482509133587102200679624457, −15.57199619197724014449153119663, −14.80400037587668626167812516783, −14.060273704265685227276295302439, −13.36884701794983302387558229128, −12.539540078258850498054574051223, −11.28609638675936164629111408097, −10.88940796762310086935708068911, −10.44241987459182784921898536650, −9.18484812832596769815688139504, −8.12786268314858761269349387396, −7.72749161206985124787973955002, −6.65537726682260892669926120745, −6.088648957562359995108673413614, −4.71052242403049992105406248394, −4.123032040492788755372275559403, −3.09455786051824967482501300702, −2.22982206977836277373284346039, −0.936404014264044621807230953352, 0.66732223634310904580152948770, 1.897133224314289440830534054333, 2.74369783386844453654763136561, 4.03105250233052712930803414216, 4.812364845204946219917074530445, 5.450069611090399278912394472655, 6.42495434450277785527567527257, 7.57740367756346046055810142154, 8.40198483562864405317033225106, 8.77194680923759998716345158139, 9.724398366461936385278665587467, 11.05666561514466285447310708716, 11.26529130511801798832263743701, 12.40572020824272140892390251327, 13.04068706706542229714689670436, 13.59591520652989613727133853229, 14.936273840545622936009022377381, 15.51222787784485745814097774552, 15.91282777418797075921653890881, 17.04411022777358242337523872595, 17.69717580597717206072937844315, 18.43627503014377578275634397543, 19.21798219753300901451427799170, 20.0497621838953382594988817167, 20.901201779619689321586781449994

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