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Properties

Label 1-2675-2675.47-r1-0-0
Degree $1$
Conductor $2675$
Sign $0.169 + 0.985i$
Analytic cond. $287.468$
Root an. cond. $287.468$
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  + (−0.985 + 0.171i)2-s + (−0.976 − 0.217i)3-s + (0.941 − 0.337i)4-s + (0.998 + 0.0474i)6-s + (0.989 + 0.147i)7-s + (−0.869 + 0.493i)8-s + (0.905 + 0.424i)9-s + (−0.297 + 0.954i)11-s + (−0.992 + 0.124i)12-s + (0.670 − 0.741i)13-s + (−0.999 + 0.0237i)14-s + (0.772 − 0.634i)16-s + (0.783 + 0.620i)17-s + (−0.964 − 0.263i)18-s + (0.666 − 0.745i)19-s + ⋯
L(s)  = 1  + (−0.985 + 0.171i)2-s + (−0.976 − 0.217i)3-s + (0.941 − 0.337i)4-s + (0.998 + 0.0474i)6-s + (0.989 + 0.147i)7-s + (−0.869 + 0.493i)8-s + (0.905 + 0.424i)9-s + (−0.297 + 0.954i)11-s + (−0.992 + 0.124i)12-s + (0.670 − 0.741i)13-s + (−0.999 + 0.0237i)14-s + (0.772 − 0.634i)16-s + (0.783 + 0.620i)17-s + (−0.964 − 0.263i)18-s + (0.666 − 0.745i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2675\)    =    \(5^{2} \cdot 107\)
Sign: $0.169 + 0.985i$
Analytic conductor: \(287.468\)
Root analytic conductor: \(287.468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2675} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2675,\ (1:\ ),\ 0.169 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7629455018 + 0.6430696120i\)
\(L(\frac12)\) \(\approx\) \(0.7629455018 + 0.6430696120i\)
\(L(1)\) \(\approx\) \(0.6268212200 + 0.06533348494i\)
\(L(1)\) \(\approx\) \(0.6268212200 + 0.06533348494i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
107 \( 1 \)
good2 \( 1 + (-0.985 + 0.171i)T \)
3 \( 1 + (-0.976 - 0.217i)T \)
7 \( 1 + (0.989 + 0.147i)T \)
11 \( 1 + (-0.297 + 0.954i)T \)
13 \( 1 + (0.670 - 0.741i)T \)
17 \( 1 + (0.783 + 0.620i)T \)
19 \( 1 + (0.666 - 0.745i)T \)
23 \( 1 + (0.958 - 0.286i)T \)
29 \( 1 + (-0.709 - 0.705i)T \)
31 \( 1 + (-0.765 - 0.643i)T \)
37 \( 1 + (-0.370 + 0.928i)T \)
41 \( 1 + (-0.342 + 0.939i)T \)
43 \( 1 + (-0.996 + 0.0887i)T \)
47 \( 1 + (-0.961 - 0.275i)T \)
53 \( 1 + (-0.467 + 0.884i)T \)
59 \( 1 + (0.988 - 0.153i)T \)
61 \( 1 + (0.461 - 0.886i)T \)
67 \( 1 + (0.413 + 0.910i)T \)
71 \( 1 + (-0.749 - 0.661i)T \)
73 \( 1 + (0.983 + 0.182i)T \)
79 \( 1 + (-0.194 - 0.980i)T \)
83 \( 1 + (0.999 + 0.0177i)T \)
89 \( 1 + (0.780 - 0.625i)T \)
97 \( 1 + (-0.518 + 0.854i)T \)
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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

−18.6389817555777655986489965527, −18.31319886775230856665035334133, −17.66848494067160152254898097454, −16.77322178628974597705768076058, −16.39221658721659428500317912069, −15.88249061917635404453111103797, −14.857274389908063584257123993689, −14.1128404449150388855871393993, −13.08752588212208952893263209893, −12.14876400493712025901954284719, −11.52238657495899543500411077137, −11.05434889948964836609198875704, −10.520581128776791046804833158912, −9.60814382762122423427826634007, −8.886471353823849240642531510022, −8.10640470061469458665040453849, −7.26198919842262752593991094851, −6.70575012986252902984250777589, −5.481875932775319366461333349898, −5.31406063025228023988225058529, −3.849705950038239936127960658781, −3.26219548697946610852854322779, −1.76769511282508255221138730445, −1.2439729202777449937233536305, −0.34752750433800734747972264780, 0.80733056744412926528649770230, 1.459043064683597849382809037853, 2.253634680745883608017429494247, 3.4735273098703150104440957825, 4.866452227645426147837146479, 5.28218094176984065155533886662, 6.15571406126783663155105662898, 6.944530390041732427243330966510, 7.75324251697838232969977496425, 8.11641088853429396714154394470, 9.228598830707325494281741323699, 10.04659196578273685995648185358, 10.62064488396832765940727637175, 11.38349206362606347262113709039, 11.76918800528972139551290897498, 12.74131589292382093586489202216, 13.38234353088074311978812557502, 14.83533487045049397516997889139, 15.076886336570487157908560855397, 15.909858038749112714936033054879, 16.71248050566429102918412018986, 17.39877049428869848854808142156, 17.72021699219858972536244651934, 18.57252794213273663656129978318, 18.78044943699007706562333279545

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