↑

Properties

Label 1-1339-1339.571-r0-0-0
Degree $1$
Conductor $1339$
Sign $0.957 + 0.287i$
Analytic cond. $6.21828$
Root an. cond. $6.21828$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + 9-s + 10-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + 14-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)18-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + 9-s + 10-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + 14-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1339\)    =    \(13 \cdot 103\)
Sign: $0.957 + 0.287i$
Analytic conductor: \(6.21828\)
Root analytic conductor: \(6.21828\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1339} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1339,\ (0:\ ),\ 0.957 + 0.287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.283739779 + 0.4826906949i\)
\(L(\frac12)\) \(\approx\) \(3.283739779 + 0.4826906949i\)
\(L(1)\) \(\approx\) \(2.022601611 + 0.4504044037i\)
\(L(1)\) \(\approx\) \(2.022601611 + 0.4504044037i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - T \)
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 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 - T \)
97 \( 1 + (0.5 + 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.6787949263433853087215823620, −20.406046166975357128173367184142, −19.31790836579941194044461423837, −18.86217803672695185526744008865, −17.964901255167683332780810229069, −17.6707385389524514980945059503, −15.83586051440989612302013437432, −15.177202872835181177868274249731, −14.51351953855872837013679835533, −14.150280450216519041267085269433, −13.153027360868151769868460477881, −12.576479357760491430949787918435, −11.4797810828680267150373368513, −10.95187486682958893936747551345, −9.841344471164828272272888053556, −9.28422389768764361655159614977, −8.74614670876803607436610772011, −7.281136228591145020252135677916, −6.73738290777977525800940816923, −5.3964976386463109775514835651, −4.75817902837033435297355357929, −3.617227154807681688727069711329, −2.81814934843525343715521306293, −2.194617992199229872455072876436, −1.46119869894787345227388861018, 1.03838310243902703310602891933, 2.07044669304821057187941725080, 3.51469398634651064644144513503, 3.96103937677174424541278993492, 4.89337479730704939204429786079, 5.77541708274224164738681576259, 6.75339515849321643083134417396, 7.63467850548022418903324263781, 8.358262612573038644509422543836, 8.90180869943317479425929155146, 9.70473908242536727098025400511, 10.79221064276544503455594470161, 11.99356081799472966600207549079, 12.91411986455050592718018088589, 13.49766591962254802386588326319, 13.99639469293825053795275565719, 14.7135674637722283967128583513, 15.47410835327551318379995104264, 16.528098161508028076022409389011, 16.84121147964357957238827869598, 17.68182555604864549988027007429, 18.61468078903957572729766066484, 19.56447402179245894718412238965, 20.42055229022535408210025476293, 20.97563668652627895352454393199

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