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Properties

Label 1-756-756.299-r1-0-0
Degree $1$
Conductor $756$
Sign $-0.163 + 0.986i$
Analytic cond. $81.2434$
Root an. cond. $81.2434$
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  + (−0.939 + 0.342i)5-s + (−0.939 − 0.342i)11-s + (0.939 − 0.342i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.173 − 0.984i)23-s + (0.766 − 0.642i)25-s + (0.939 + 0.342i)29-s + (−0.939 + 0.342i)31-s + 37-s + (−0.939 + 0.342i)41-s + (−0.173 − 0.984i)43-s + (0.939 + 0.342i)47-s + (0.5 − 0.866i)53-s + 55-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)5-s + (−0.939 − 0.342i)11-s + (0.939 − 0.342i)13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.173 − 0.984i)23-s + (0.766 − 0.642i)25-s + (0.939 + 0.342i)29-s + (−0.939 + 0.342i)31-s + 37-s + (−0.939 + 0.342i)41-s + (−0.173 − 0.984i)43-s + (0.939 + 0.342i)47-s + (0.5 − 0.866i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.163 + 0.986i$
Analytic conductor: \(81.2434\)
Root analytic conductor: \(81.2434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 756,\ (1:\ ),\ -0.163 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4880029368 + 0.5754781415i\)
\(L(\frac12)\) \(\approx\) \(0.4880029368 + 0.5754781415i\)
\(L(1)\) \(\approx\) \(0.7912388489 + 0.03663778720i\)
\(L(1)\) \(\approx\) \(0.7912388489 + 0.03663778720i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (-0.939 - 0.342i)T \)
13 \( 1 + (0.939 - 0.342i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.766 - 0.642i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (0.939 + 0.342i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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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

−21.87190226399180374316304082046, −21.22408574744483094479906325999, −20.2731014231306209649576553327, −19.67027110178365718439825405829, −18.827132395468674331675060025399, −18.02591276942565946732640373017, −17.10174471374940000961940171968, −16.178164559066921839135813385278, −15.44732513743794828234277632197, −14.98353038712189547591368621925, −13.53776778452362176548198795054, −13.036587363334790301927097782309, −12.08182405134819840814828650684, −11.16474736637476684408361155367, −10.59253850294086456705694269818, −9.29469416204913647875912042475, −8.49151776001076732377107658424, −7.74900607112552392508403902623, −6.82010626435052192953161477154, −5.73458037697269332293174972468, −4.63355264550217558368271636362, −3.93281039729262744000308784305, −2.81256020338470616941649623269, −1.531995633932126078286124830339, −0.22602564175764061315993972827, 0.83637348738956730870688820780, 2.442217003992968939802612343442, 3.33037801925841024601179380540, 4.26896068144137420863717212525, 5.297791418477706680782441814434, 6.39102244630405349854633994375, 7.27986948106922049282243886978, 8.23770099019239821597005153162, 8.75427837111765096385020977672, 10.251286835414716305431742742439, 10.80309186964063925693393738105, 11.6301019428064434543582938519, 12.56795388072828620033496195037, 13.36597398175987150680279606130, 14.36013392584733140600931574513, 15.180408524959420097848151185509, 16.040341325686013335249572998049, 16.41878363152162563067177532526, 17.83513880630101627064864055878, 18.53583532517472705604248622055, 19.01242894096005713371654651036, 20.26150344528834130213361079561, 20.57212742552005581595540031753, 21.71665364335142139304875285659, 22.53678565818543455096390198564

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