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Properties

Label 1-269-269.127-r0-0-0
Degree $1$
Conductor $269$
Sign $0.982 + 0.185i$
Analytic cond. $1.24923$
Root an. cond. $1.24923$
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  + (−0.430 − 0.902i)2-s + (0.990 + 0.140i)3-s + (−0.628 + 0.777i)4-s + (−0.0234 + 0.999i)5-s + (−0.300 − 0.953i)6-s + (0.819 − 0.572i)7-s + (0.972 + 0.232i)8-s + (0.960 + 0.277i)9-s + (0.912 − 0.409i)10-s + (0.344 + 0.938i)11-s + (−0.731 + 0.681i)12-s + (−0.912 + 0.409i)13-s + (−0.869 − 0.493i)14-s + (−0.163 + 0.986i)15-s + (−0.209 − 0.977i)16-s + (−0.344 + 0.938i)17-s + ⋯
L(s)  = 1  + (−0.430 − 0.902i)2-s + (0.990 + 0.140i)3-s + (−0.628 + 0.777i)4-s + (−0.0234 + 0.999i)5-s + (−0.300 − 0.953i)6-s + (0.819 − 0.572i)7-s + (0.972 + 0.232i)8-s + (0.960 + 0.277i)9-s + (0.912 − 0.409i)10-s + (0.344 + 0.938i)11-s + (−0.731 + 0.681i)12-s + (−0.912 + 0.409i)13-s + (−0.869 − 0.493i)14-s + (−0.163 + 0.986i)15-s + (−0.209 − 0.977i)16-s + (−0.344 + 0.938i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(269\)
Sign: $0.982 + 0.185i$
Analytic conductor: \(1.24923\)
Root analytic conductor: \(1.24923\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{269} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 269,\ (0:\ ),\ 0.982 + 0.185i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.352102338 + 0.1264228988i\)
\(L(\frac12)\) \(\approx\) \(1.352102338 + 0.1264228988i\)
\(L(1)\) \(\approx\) \(1.168419917 - 0.08184706791i\)
\(L(1)\) \(\approx\) \(1.168419917 - 0.08184706791i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 \)
good2 \( 1 + (-0.430 - 0.902i)T \)
3 \( 1 + (0.990 + 0.140i)T \)
5 \( 1 + (-0.0234 + 0.999i)T \)
7 \( 1 + (0.819 - 0.572i)T \)
11 \( 1 + (0.344 + 0.938i)T \)
13 \( 1 + (-0.912 + 0.409i)T \)
17 \( 1 + (-0.344 + 0.938i)T \)
19 \( 1 + (-0.930 + 0.366i)T \)
23 \( 1 + (-0.990 - 0.140i)T \)
29 \( 1 + (0.998 - 0.0468i)T \)
31 \( 1 + (0.762 - 0.646i)T \)
37 \( 1 + (0.982 + 0.186i)T \)
41 \( 1 + (0.430 - 0.902i)T \)
43 \( 1 + (-0.998 + 0.0468i)T \)
47 \( 1 + (-0.116 - 0.993i)T \)
53 \( 1 + (0.792 + 0.610i)T \)
59 \( 1 + (0.628 - 0.777i)T \)
61 \( 1 + (0.591 + 0.806i)T \)
67 \( 1 + (-0.628 - 0.777i)T \)
71 \( 1 + (0.869 - 0.493i)T \)
73 \( 1 + (0.513 + 0.858i)T \)
79 \( 1 + (-0.698 + 0.715i)T \)
83 \( 1 + (-0.845 - 0.533i)T \)
89 \( 1 + (-0.553 - 0.833i)T \)
97 \( 1 + (0.591 - 0.806i)T \)
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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

−25.40703775321823852456906411528, −24.83151547096297486597639702052, −24.330942429474181891818367617453, −23.56883485638160223837875382940, −21.95308249472855942864232914682, −21.166136652013872168989593311386, −19.90940883215151655413185933122, −19.43968802637074232886117700324, −18.234417529464982844283656741095, −17.49254602048440609886872783054, −16.33060557660384910913921901195, −15.56645880244507327249531393327, −14.60654685480896162015029024899, −13.86527738144083962115680104877, −12.89066495934289764571635031931, −11.665981839489453992001489638210, −10.04736278368044456185789664368, −9.07442723715851763364884247051, −8.40544327668001995115838463938, −7.794193088536400574818187301921, −6.42042553843865283587398961303, −5.110265123455187807377347002647, −4.31653033878708701276177145123, −2.437959937311198288259938540395, −1.05114632685767641725269596338, 1.83682383745513192702038419884, 2.4287891129726853891212459122, 3.93912716315312791126276645135, 4.44089935890591005862591118060, 6.81724953919887994658492643061, 7.741959718252249879276852184766, 8.5348966756488759661284368730, 9.99580697913302626269125066457, 10.24181279284666719656467390738, 11.499708557884096707014398302233, 12.53798006042568644378909165363, 13.75157307466666331022499274030, 14.490122704801161291440252919210, 15.20174548950051563755472348842, 16.90927649468822723223204939895, 17.7294073941317847233911825220, 18.63893701163527202416162404792, 19.62007689658967032725001447226, 20.04331730932089864382670897328, 21.23600887558001147416404405503, 21.74900766212094363891925641194, 22.828629588926231654934185238010, 23.9631338018669100569782934920, 25.27742309688900504850486062107, 26.074481942248766740459078081953

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