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Properties

Label 1-1275-1275.314-r1-0-0
Degree $1$
Conductor $1275$
Sign $-0.623 + 0.782i$
Analytic cond. $137.017$
Root an. cond. $137.017$
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  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.707 + 0.707i)7-s + (−0.587 − 0.809i)8-s + (−0.453 − 0.891i)11-s + (0.309 + 0.951i)13-s + (0.891 − 0.453i)14-s + (0.309 + 0.951i)16-s + (0.587 + 0.809i)19-s + (0.156 + 0.987i)22-s + (0.453 + 0.891i)23-s − i·26-s + (−0.987 + 0.156i)28-s + (0.156 + 0.987i)29-s + (0.987 + 0.156i)31-s − i·32-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.707 + 0.707i)7-s + (−0.587 − 0.809i)8-s + (−0.453 − 0.891i)11-s + (0.309 + 0.951i)13-s + (0.891 − 0.453i)14-s + (0.309 + 0.951i)16-s + (0.587 + 0.809i)19-s + (0.156 + 0.987i)22-s + (0.453 + 0.891i)23-s − i·26-s + (−0.987 + 0.156i)28-s + (0.156 + 0.987i)29-s + (0.987 + 0.156i)31-s − i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.623 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.623 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1275\)    =    \(3 \cdot 5^{2} \cdot 17\)
Sign: $-0.623 + 0.782i$
Analytic conductor: \(137.017\)
Root analytic conductor: \(137.017\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1275} (314, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1275,\ (1:\ ),\ -0.623 + 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3304267006 + 0.6858426584i\)
\(L(\frac12)\) \(\approx\) \(0.3304267006 + 0.6858426584i\)
\(L(1)\) \(\approx\) \(0.6365515330 + 0.08363675097i\)
\(L(1)\) \(\approx\) \(0.6365515330 + 0.08363675097i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (-0.453 - 0.891i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.587 + 0.809i)T \)
23 \( 1 + (0.453 + 0.891i)T \)
29 \( 1 + (0.156 + 0.987i)T \)
31 \( 1 + (0.987 + 0.156i)T \)
37 \( 1 + (-0.453 + 0.891i)T \)
41 \( 1 + (-0.891 - 0.453i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.587 + 0.809i)T \)
59 \( 1 + (0.951 - 0.309i)T \)
61 \( 1 + (-0.453 - 0.891i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (-0.156 - 0.987i)T \)
73 \( 1 + (-0.891 + 0.453i)T \)
79 \( 1 + (0.987 - 0.156i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.156 + 0.987i)T \)
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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

−20.49840163125820671055760961627, −19.68761305795566086134086597773, −19.09541534536061628549176936295, −18.08128616407510503707554600093, −17.60420033938577414925738176431, −16.85145961583971895569105976251, −15.973473202086054481378688063276, −15.46326504409621856391039323603, −14.67912601482537518767435746209, −13.567666132925787699932561796091, −12.87249431853589613365850537131, −11.8862330361643443940974431580, −10.910347147298798754917646473366, −10.171146776851143027285262828, −9.74234819709888141328644737615, −8.70335316948957799096280415482, −7.853545417225545382931935540672, −7.13028467410964035058908727953, −6.45653798306474305562570198058, −5.45982835830627112202096451085, −4.46753586502912491410792286147, −3.13072074434403582078744723777, −2.36688237877119535917553667493, −0.99026062934120898179484182912, −0.269342390330543365983622649, 1.00845803787959340194492603699, 2.00476846084908416476018768642, 3.09629981292982913778328708625, 3.603818395548661765688544178217, 5.18902124540118257896791168841, 6.14528965745398343108187829078, 6.82923271500328960864501876897, 7.85028323880244228656035148491, 8.69998972207041178540753954453, 9.2363055792706906114940382816, 10.0966178785585258308180925389, 10.88033060673828899769894012086, 11.78860063650325178998180444834, 12.266299021831806836009835972164, 13.31652907404799948321197971459, 14.07485468904170423825761793580, 15.415653494174942987707059259596, 15.86228951999176375972662119669, 16.575629131151057546031949977594, 17.286072945518686272702017895, 18.372925288246695715220436149429, 18.849549871524191791935310290500, 19.262426901893592563484976210232, 20.30329923517937509386808155550, 21.03253100601461949920637292187

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