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Properties

Label 1-2672-2672.331-r0-0-0
Degree $1$
Conductor $2672$
Sign $0.609 + 0.793i$
Analytic cond. $12.4087$
Root an. cond. $12.4087$
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 + 0.132i)3-s + (−0.404 + 0.914i)5-s + (0.421 − 0.906i)7-s + (0.965 − 0.261i)9-s + (−0.788 − 0.614i)11-s + (−0.890 − 0.455i)13-s + (0.280 − 0.959i)15-s + (−0.997 − 0.0756i)17-s + (0.832 + 0.553i)19-s + (−0.298 + 0.954i)21-s + (−0.929 − 0.369i)23-s + (−0.672 − 0.739i)25-s + (−0.922 + 0.387i)27-s + (−0.369 − 0.929i)29-s + (−0.974 + 0.225i)31-s + ⋯
L(s)  = 1  + (−0.991 + 0.132i)3-s + (−0.404 + 0.914i)5-s + (0.421 − 0.906i)7-s + (0.965 − 0.261i)9-s + (−0.788 − 0.614i)11-s + (−0.890 − 0.455i)13-s + (0.280 − 0.959i)15-s + (−0.997 − 0.0756i)17-s + (0.832 + 0.553i)19-s + (−0.298 + 0.954i)21-s + (−0.929 − 0.369i)23-s + (−0.672 − 0.739i)25-s + (−0.922 + 0.387i)27-s + (−0.369 − 0.929i)29-s + (−0.974 + 0.225i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.609 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.609 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2672\)    =    \(2^{4} \cdot 167\)
Sign: $0.609 + 0.793i$
Analytic conductor: \(12.4087\)
Root analytic conductor: \(12.4087\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2672} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2672,\ (0:\ ),\ 0.609 + 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4672092900 + 0.2303112224i\)
\(L(\frac12)\) \(\approx\) \(0.4672092900 + 0.2303112224i\)
\(L(1)\) \(\approx\) \(0.5993316677 + 0.02845384992i\)
\(L(1)\) \(\approx\) \(0.5993316677 + 0.02845384992i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 \)
good3 \( 1 + (-0.991 + 0.132i)T \)
5 \( 1 + (-0.404 + 0.914i)T \)
7 \( 1 + (0.421 - 0.906i)T \)
11 \( 1 + (-0.788 - 0.614i)T \)
13 \( 1 + (-0.890 - 0.455i)T \)
17 \( 1 + (-0.997 - 0.0756i)T \)
19 \( 1 + (0.832 + 0.553i)T \)
23 \( 1 + (-0.929 - 0.369i)T \)
29 \( 1 + (-0.369 - 0.929i)T \)
31 \( 1 + (-0.974 + 0.225i)T \)
37 \( 1 + (-0.261 + 0.965i)T \)
41 \( 1 + (0.898 - 0.438i)T \)
43 \( 1 + (-0.995 - 0.0944i)T \)
47 \( 1 + (0.0189 + 0.999i)T \)
53 \( 1 + (-0.985 + 0.169i)T \)
59 \( 1 + (0.0756 + 0.997i)T \)
61 \( 1 + (0.713 + 0.700i)T \)
67 \( 1 + (0.404 + 0.914i)T \)
71 \( 1 + (0.993 - 0.113i)T \)
73 \( 1 + (-0.800 + 0.599i)T \)
79 \( 1 + (0.351 - 0.936i)T \)
83 \( 1 + (0.811 + 0.584i)T \)
89 \( 1 + (-0.280 - 0.959i)T \)
97 \( 1 + (0.974 + 0.225i)T \)
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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

−19.17712833467043387700660019034, −18.11132102629882125018609140353, −17.98683761431621048412643802095, −17.13019173036338506984152441792, −16.33155148235251340913678348188, −15.75845466152482299017495888248, −15.26884262338253281671750051991, −14.28086584994397641167056763780, −13.13939048746689273094153320554, −12.677294047205146957925971195006, −12.0578399600545445721040424267, −11.447056813322006816730643197836, −10.82943102868984926223955154315, −9.62312210612866489923921461279, −9.27777799655509996305618676418, −8.17237197449809155502089927400, −7.518542783680585903735913993374, −6.78000017477641109885520874, −5.68503006289752158668613367444, −5.02788365078142345108148844081, −4.763975700096830392515127839682, −3.68160156006833548414977915564, −2.176576821742030657622993135570, −1.76150658930446249057639698309, −0.31663220499916039042363086049, 0.585280533872731053511645182735, 1.88808867563550254724580960339, 2.92948841506897729211497093229, 3.87116181201370083658574406715, 4.53964542957973469156726205532, 5.42388866729765370588941566766, 6.15769521736703035208017406863, 7.05176277223323803467335698349, 7.56870256619557708855307072647, 8.22624987212633386380584495065, 9.685322531067635023439309953550, 10.24884494868018038537054582464, 10.83040355706640504815774203905, 11.39454795715731015290204111950, 12.0526017120867646106302563875, 12.99087636044104985006613752441, 13.73041846187653600241367543496, 14.48974242633332800190437851860, 15.27344653807045297873851394684, 15.97464177501992949107836100120, 16.53021119827148778505070553430, 17.48847041741877385213257078404, 17.87875256301636629506626376101, 18.56764375875623881996367164687, 19.27527938294214055775804301189

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