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Properties

Label 1-269-269.16-r0-0-0
Degree $1$
Conductor $269$
Sign $0.771 + 0.635i$
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  + (0.995 + 0.0936i)2-s + (−0.698 − 0.715i)3-s + (0.982 + 0.186i)4-s + (0.792 + 0.610i)5-s + (−0.628 − 0.777i)6-s + (−0.209 + 0.977i)7-s + (0.960 + 0.277i)8-s + (−0.0234 + 0.999i)9-s + (0.731 + 0.681i)10-s + (−0.912 + 0.409i)11-s + (−0.553 − 0.833i)12-s + (0.731 + 0.681i)13-s + (−0.300 + 0.953i)14-s + (−0.116 − 0.993i)15-s + (0.930 + 0.366i)16-s + (−0.912 − 0.409i)17-s + ⋯
L(s)  = 1  + (0.995 + 0.0936i)2-s + (−0.698 − 0.715i)3-s + (0.982 + 0.186i)4-s + (0.792 + 0.610i)5-s + (−0.628 − 0.777i)6-s + (−0.209 + 0.977i)7-s + (0.960 + 0.277i)8-s + (−0.0234 + 0.999i)9-s + (0.731 + 0.681i)10-s + (−0.912 + 0.409i)11-s + (−0.553 − 0.833i)12-s + (0.731 + 0.681i)13-s + (−0.300 + 0.953i)14-s + (−0.116 − 0.993i)15-s + (0.930 + 0.366i)16-s + (−0.912 − 0.409i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.771 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.771 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(269\)
Sign: $0.771 + 0.635i$
Analytic conductor: \(1.24923\)
Root analytic conductor: \(1.24923\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{269} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 269,\ (0:\ ),\ 0.771 + 0.635i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.842275517 + 0.6612353294i\)
\(L(\frac12)\) \(\approx\) \(1.842275517 + 0.6612353294i\)
\(L(1)\) \(\approx\) \(1.625478835 + 0.2520832124i\)
\(L(1)\) \(\approx\) \(1.625478835 + 0.2520832124i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 \)
good2 \( 1 + (0.995 + 0.0936i)T \)
3 \( 1 + (-0.698 - 0.715i)T \)
5 \( 1 + (0.792 + 0.610i)T \)
7 \( 1 + (-0.209 + 0.977i)T \)
11 \( 1 + (-0.912 + 0.409i)T \)
13 \( 1 + (0.731 + 0.681i)T \)
17 \( 1 + (-0.912 - 0.409i)T \)
19 \( 1 + (-0.472 + 0.881i)T \)
23 \( 1 + (-0.698 - 0.715i)T \)
29 \( 1 + (0.255 - 0.966i)T \)
31 \( 1 + (0.664 - 0.747i)T \)
37 \( 1 + (0.513 - 0.858i)T \)
41 \( 1 + (0.995 - 0.0936i)T \)
43 \( 1 + (0.255 - 0.966i)T \)
47 \( 1 + (-0.990 + 0.140i)T \)
53 \( 1 + (0.892 - 0.451i)T \)
59 \( 1 + (0.982 + 0.186i)T \)
61 \( 1 + (0.430 + 0.902i)T \)
67 \( 1 + (0.982 - 0.186i)T \)
71 \( 1 + (-0.300 - 0.953i)T \)
73 \( 1 + (-0.819 - 0.572i)T \)
79 \( 1 + (-0.946 + 0.322i)T \)
83 \( 1 + (-0.998 - 0.0468i)T \)
89 \( 1 + (-0.762 + 0.646i)T \)
97 \( 1 + (0.430 - 0.902i)T \)
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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

−25.76316162970684756767372680769, −24.383549960858587513325388312103, −23.64181409507281699233753035754, −23.00841301181976766261439002688, −21.84878623545322162014389452136, −21.3727946054778889460418222526, −20.45245853910885821034063806693, −19.80520774785340177401560304908, −17.93305531585563071345556935610, −17.207281538395654234236705975643, −16.091692260327452799183799512548, −15.7223272444980510302047914307, −14.34554060157014054161782012373, −13.20228889940608860366064781034, −12.91604238027913315711404455212, −11.35672197158519779937164517715, −10.62017127208490776931939600880, −9.91358976871716128855930530547, −8.38581390124812266456279893589, −6.7512602302223810392275623350, −5.90755442977753864206494900264, −4.98893525477386924542228832736, −4.13549860344931361234467076670, −2.897459425147839171403085897802, −1.11817793902031898623292115532, 2.066592495330084008173532739709, 2.47827631405894320970070964318, 4.34792154699780264248610395213, 5.6722760993369805858204845746, 6.13352018714841085989285946719, 7.03918750393788519279556831487, 8.30238760779954844786349621867, 10.02073293687841628900101614350, 11.027610137541374467586643853398, 11.87661562618732406741105131066, 12.872032282854130617592834911108, 13.50830744284711710998750326116, 14.50300173979832512569642706966, 15.62782567727215749422167182168, 16.441584089456406354138364283029, 17.70174806417076449354418246819, 18.43516743659849337266504586220, 19.2747891111164280463327377256, 20.84708086819358172833826014202, 21.46612309345712076293400572636, 22.54015894086826130140860154384, 22.86217024781882343749881507651, 24.03238082216927702910153758626, 24.80288264193846442140321466807, 25.54504592752072846703923070289

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