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Properties

Label 1-1224-1224.67-r1-0-0
Degree $1$
Conductor $1224$
Sign $0.766 + 0.642i$
Analytic cond. $131.537$
Root an. cond. $131.537$
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  + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s + 35-s + 37-s + (0.5 + 0.866i)41-s + (−0.5 + 0.866i)43-s + (0.5 − 0.866i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s + 35-s + 37-s + (0.5 + 0.866i)41-s + (−0.5 + 0.866i)43-s + (0.5 − 0.866i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(131.537\)
Root analytic conductor: \(131.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1224,\ (1:\ ),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.404829826 + 0.5113162410i\)
\(L(\frac12)\) \(\approx\) \(1.404829826 + 0.5113162410i\)
\(L(1)\) \(\approx\) \(0.9502007691 + 0.01481737564i\)
\(L(1)\) \(\approx\) \(0.9502007691 + 0.01481737564i\)

Euler product

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

−20.54614399995702155969725628799, −20.10568320954259743010724339479, −19.437111210444162032059321845206, −18.56352309616889309259454423236, −17.76420026082740972215547346761, −17.194073687508000180628864884574, −16.00414692766701088465467770465, −15.59048049948465579842212939580, −14.64956181644313086194252196702, −13.9426247680424098136170861382, −13.16048996933886346692331900616, −12.222762825130502784987455790190, −11.42685898415390342593677373589, −10.59672977817664513358771437865, −9.96518307056036748483266946905, −9.153574512649529520158735915076, −7.68089703449818878205741050263, −7.49850798802063209470387642391, −6.514573984035679601657150707406, −5.66951938466634232006934759555, −4.35913174419510947363840513063, −3.63279447138488026190317180096, −2.935941728779974032934283963244, −1.59604315742678490746471663498, −0.41425421830330092434932199132, 0.752394697994236524266454453590, 1.77344876814029296266069718941, 3.04931620078748802706328138733, 3.869362739340464852787037760656, 4.80048892421846808333558621115, 5.80221148335933868064205132732, 6.40957670593653670203615852051, 7.61792170216506832179448960286, 8.49808407471458733780407828463, 9.115195868500196284283012599527, 9.704019361698105712982553795887, 11.19031453481363533291147692880, 11.60929523509069944564134001275, 12.4599525548746805494958381174, 13.14478449643335172831227567550, 14.040344380093005719788963544366, 14.90330796388556508390228341642, 15.87718033815559998095903283732, 16.40509785170004675187753546963, 16.83117116500796808349147756816, 18.30917033548478720398494453222, 18.60691109717580728084496701570, 19.61113409545744914039884980263, 20.13073812087608135607950408805, 21.06105611456074111580610138349

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