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Properties

Label 1-269-269.100-r0-0-0
Degree $1$
Conductor $269$
Sign $-0.201 - 0.979i$
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  + (−0.930 − 0.366i)2-s + (0.998 + 0.0468i)3-s + (0.731 + 0.681i)4-s + (−0.869 + 0.493i)5-s + (−0.912 − 0.409i)6-s + (−0.664 − 0.747i)7-s + (−0.430 − 0.902i)8-s + (0.995 + 0.0936i)9-s + (0.990 − 0.140i)10-s + (−0.116 − 0.993i)11-s + (0.698 + 0.715i)12-s + (−0.990 + 0.140i)13-s + (0.344 + 0.938i)14-s + (−0.892 + 0.451i)15-s + (0.0702 + 0.997i)16-s + (0.116 − 0.993i)17-s + ⋯
L(s)  = 1  + (−0.930 − 0.366i)2-s + (0.998 + 0.0468i)3-s + (0.731 + 0.681i)4-s + (−0.869 + 0.493i)5-s + (−0.912 − 0.409i)6-s + (−0.664 − 0.747i)7-s + (−0.430 − 0.902i)8-s + (0.995 + 0.0936i)9-s + (0.990 − 0.140i)10-s + (−0.116 − 0.993i)11-s + (0.698 + 0.715i)12-s + (−0.990 + 0.140i)13-s + (0.344 + 0.938i)14-s + (−0.892 + 0.451i)15-s + (0.0702 + 0.997i)16-s + (0.116 − 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(269\)
Sign: $-0.201 - 0.979i$
Analytic conductor: \(1.24923\)
Root analytic conductor: \(1.24923\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{269} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 269,\ (0:\ ),\ -0.201 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4457190622 - 0.5469086920i\)
\(L(\frac12)\) \(\approx\) \(0.4457190622 - 0.5469086920i\)
\(L(1)\) \(\approx\) \(0.6924567841 - 0.2433872075i\)
\(L(1)\) \(\approx\) \(0.6924567841 - 0.2433872075i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 \)
good2 \( 1 + (-0.930 - 0.366i)T \)
3 \( 1 + (0.998 + 0.0468i)T \)
5 \( 1 + (-0.869 + 0.493i)T \)
7 \( 1 + (-0.664 - 0.747i)T \)
11 \( 1 + (-0.116 - 0.993i)T \)
13 \( 1 + (-0.990 + 0.140i)T \)
17 \( 1 + (0.116 - 0.993i)T \)
19 \( 1 + (0.388 - 0.921i)T \)
23 \( 1 + (-0.998 - 0.0468i)T \)
29 \( 1 + (-0.513 - 0.858i)T \)
31 \( 1 + (0.972 - 0.232i)T \)
37 \( 1 + (-0.553 + 0.833i)T \)
41 \( 1 + (0.930 - 0.366i)T \)
43 \( 1 + (0.513 + 0.858i)T \)
47 \( 1 + (0.845 - 0.533i)T \)
53 \( 1 + (-0.300 - 0.953i)T \)
59 \( 1 + (-0.731 - 0.681i)T \)
61 \( 1 + (-0.209 - 0.977i)T \)
67 \( 1 + (0.731 - 0.681i)T \)
71 \( 1 + (-0.344 + 0.938i)T \)
73 \( 1 + (-0.762 + 0.646i)T \)
79 \( 1 + (0.255 - 0.966i)T \)
83 \( 1 + (-0.982 - 0.186i)T \)
89 \( 1 + (-0.946 - 0.322i)T \)
97 \( 1 + (-0.209 + 0.977i)T \)
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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

−26.03718544126127501140771066076, −25.20323475356236737202469957088, −24.52829602793886042284335702957, −23.69384083353360996876116526322, −22.46705362101313867144285919722, −21.07581321124360299097197891177, −20.13434760777946530682793315238, −19.55357806876715153205752522268, −18.90143014280555711975808551222, −17.885833048066433216747132121425, −16.64444189435161455970745866366, −15.70139626867339613993485831174, −15.17262275667509274169194788063, −14.29333952104515066688667501730, −12.47905743828458032906160626185, −12.236904627862222080134244911412, −10.40408955220487671799225618337, −9.57717080770816531826312874888, −8.75137867350021498860207791870, −7.81637311751428702803546384361, −7.16524851108530546235338888508, −5.692582667695317545680014136546, −4.2067526381488442597429564556, −2.804394607439573412285477492481, −1.65071971605431848266748417182, 0.57087210809399637586992668486, 2.55244879967137629213232161846, 3.23583195219895613393597825037, 4.277875639162278533389607896, 6.63555373276314380948273353289, 7.46257048236388886057324423991, 8.12278399138405383284011914730, 9.36164417833823768221837914675, 10.064164254184048549100125666421, 11.14920399541411405639095074681, 12.11810759354169108732902190976, 13.36331994710079444916699867583, 14.28519008139955164471265741945, 15.670571442175490289845181100698, 16.04194437129514782031556333955, 17.258587715711424681838227238992, 18.605030285230931980084397357894, 19.17230621835743989410960529093, 19.84975015551192780463678542992, 20.5015320978483883858132602608, 21.69724885466581749325986441069, 22.57421950490268408804589659814, 24.0804474864810433710597801871, 24.67346148305308428618328456002, 26.07644407953629104724771960659

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