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Properties

Label 1-612-612.79-r0-0-0
Degree $1$
Conductor $612$
Sign $0.731 + 0.682i$
Analytic cond. $2.84211$
Root an. cond. $2.84211$
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  + (−0.991 − 0.130i)5-s + (0.991 − 0.130i)7-s + (0.130 + 0.991i)11-s + (0.866 + 0.5i)13-s + (−0.707 − 0.707i)19-s + (−0.793 + 0.608i)23-s + (0.965 + 0.258i)25-s + (−0.608 + 0.793i)29-s + (0.130 − 0.991i)31-s − 35-s + (−0.923 + 0.382i)37-s + (0.608 + 0.793i)41-s + (0.965 + 0.258i)43-s + (0.866 − 0.5i)47-s + (0.965 − 0.258i)49-s + ⋯
L(s)  = 1  + (−0.991 − 0.130i)5-s + (0.991 − 0.130i)7-s + (0.130 + 0.991i)11-s + (0.866 + 0.5i)13-s + (−0.707 − 0.707i)19-s + (−0.793 + 0.608i)23-s + (0.965 + 0.258i)25-s + (−0.608 + 0.793i)29-s + (0.130 − 0.991i)31-s − 35-s + (−0.923 + 0.382i)37-s + (0.608 + 0.793i)41-s + (0.965 + 0.258i)43-s + (0.866 − 0.5i)47-s + (0.965 − 0.258i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(612\)    =    \(2^{2} \cdot 3^{2} \cdot 17\)
Sign: $0.731 + 0.682i$
Analytic conductor: \(2.84211\)
Root analytic conductor: \(2.84211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{612} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 612,\ (0:\ ),\ 0.731 + 0.682i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.157958779 + 0.4561860289i\)
\(L(\frac12)\) \(\approx\) \(1.157958779 + 0.4561860289i\)
\(L(1)\) \(\approx\) \(1.014107442 + 0.1200636525i\)
\(L(1)\) \(\approx\) \(1.014107442 + 0.1200636525i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 \)
good5 \( 1 + (-0.991 - 0.130i)T \)
7 \( 1 + (0.991 - 0.130i)T \)
11 \( 1 + (0.130 + 0.991i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (-0.793 + 0.608i)T \)
29 \( 1 + (-0.608 + 0.793i)T \)
31 \( 1 + (0.130 - 0.991i)T \)
37 \( 1 + (-0.923 + 0.382i)T \)
41 \( 1 + (0.608 + 0.793i)T \)
43 \( 1 + (0.965 + 0.258i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (0.258 + 0.965i)T \)
61 \( 1 + (0.991 - 0.130i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.923 - 0.382i)T \)
73 \( 1 + (0.382 + 0.923i)T \)
79 \( 1 + (0.130 + 0.991i)T \)
83 \( 1 + (0.258 - 0.965i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.608 + 0.793i)T \)
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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

−22.9686589516401639015936763332, −22.239009345137273450649862211141, −21.06548529150962919228383013482, −20.660825521034496720698843097179, −19.5209293604229767870823833634, −18.845183585646875348566924282646, −18.09548494438602629084711550453, −17.11947473223142951019289269392, −16.11180586602856078337397666262, −15.52246754920121287702415394174, −14.51109334879636152188531300366, −13.908321757208740952683681619985, −12.64081722507099774152719299867, −11.86477001328745484861615771336, −10.9809315817900837368686765399, −10.51467402466623802876096062941, −8.82151886505483750160019017085, −8.30227057764041638273263812453, −7.56709474849531557848634855878, −6.30111827791175365157655546169, −5.39847467431935449006543222576, −4.15991170598263976235933884826, −3.52386050968663845899389107318, −2.13887807326662697934562568850, −0.74341677202602041719778721070, 1.201662890284246624922621034526, 2.32131317047001371605370940519, 3.918485828390667399154802328302, 4.3574641090536322011919936789, 5.4697260488533581726506296389, 6.82440339205062167995337914207, 7.58614808076919182269517504773, 8.427531915242938628249883928364, 9.257616474710657661518180262301, 10.570799757379876347625465727007, 11.33316310559563369813951554920, 11.97630792409257943736608886729, 12.952649880453800013546371718516, 13.982734164557816483249970068656, 14.93695085497302371016804102011, 15.4736362949867642203978831600, 16.455119855324370742646215015322, 17.37759124230796157976899402988, 18.15993484046949725294861166481, 19.023602325929714415665474702823, 19.97563035930774452714172877404, 20.55523058665269266099574044301, 21.38033677255559983197718463655, 22.422691972666761340226075556723, 23.328460742387345171055211773767

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