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Properties

Label 1-269-269.5-r0-0-0
Degree $1$
Conductor $269$
Sign $0.989 + 0.144i$
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  + (0.163 − 0.986i)2-s + (−0.819 − 0.572i)3-s + (−0.946 − 0.322i)4-s + (−0.912 + 0.409i)5-s + (−0.698 + 0.715i)6-s + (−0.0234 − 0.999i)7-s + (−0.472 + 0.881i)8-s + (0.344 + 0.938i)9-s + (0.255 + 0.966i)10-s + (−0.998 − 0.0468i)11-s + (0.591 + 0.806i)12-s + (0.255 + 0.966i)13-s + (−0.990 − 0.140i)14-s + (0.982 + 0.186i)15-s + (0.792 + 0.610i)16-s + (−0.998 + 0.0468i)17-s + ⋯
L(s)  = 1  + (0.163 − 0.986i)2-s + (−0.819 − 0.572i)3-s + (−0.946 − 0.322i)4-s + (−0.912 + 0.409i)5-s + (−0.698 + 0.715i)6-s + (−0.0234 − 0.999i)7-s + (−0.472 + 0.881i)8-s + (0.344 + 0.938i)9-s + (0.255 + 0.966i)10-s + (−0.998 − 0.0468i)11-s + (0.591 + 0.806i)12-s + (0.255 + 0.966i)13-s + (−0.990 − 0.140i)14-s + (0.982 + 0.186i)15-s + (0.792 + 0.610i)16-s + (−0.998 + 0.0468i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(269\)
Sign: $0.989 + 0.144i$
Analytic conductor: \(1.24923\)
Root analytic conductor: \(1.24923\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{269} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 269,\ (0:\ ),\ 0.989 + 0.144i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3685977511 + 0.02683173975i\)
\(L(\frac12)\) \(\approx\) \(0.3685977511 + 0.02683173975i\)
\(L(1)\) \(\approx\) \(0.4928081763 - 0.2730934319i\)
\(L(1)\) \(\approx\) \(0.4928081763 - 0.2730934319i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 \)
good2 \( 1 + (0.163 - 0.986i)T \)
3 \( 1 + (-0.819 - 0.572i)T \)
5 \( 1 + (-0.912 + 0.409i)T \)
7 \( 1 + (-0.0234 - 0.999i)T \)
11 \( 1 + (-0.998 - 0.0468i)T \)
13 \( 1 + (0.255 + 0.966i)T \)
17 \( 1 + (-0.998 + 0.0468i)T \)
19 \( 1 + (0.892 + 0.451i)T \)
23 \( 1 + (-0.819 - 0.572i)T \)
29 \( 1 + (0.664 + 0.747i)T \)
31 \( 1 + (0.995 + 0.0936i)T \)
37 \( 1 + (-0.972 + 0.232i)T \)
41 \( 1 + (0.163 + 0.986i)T \)
43 \( 1 + (0.664 + 0.747i)T \)
47 \( 1 + (0.513 - 0.858i)T \)
53 \( 1 + (0.731 + 0.681i)T \)
59 \( 1 + (-0.946 - 0.322i)T \)
61 \( 1 + (-0.388 + 0.921i)T \)
67 \( 1 + (-0.946 + 0.322i)T \)
71 \( 1 + (-0.990 + 0.140i)T \)
73 \( 1 + (0.960 + 0.277i)T \)
79 \( 1 + (-0.209 + 0.977i)T \)
83 \( 1 + (-0.762 + 0.646i)T \)
89 \( 1 + (0.430 + 0.902i)T \)
97 \( 1 + (-0.388 - 0.921i)T \)
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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

−25.81365781454183466553829700421, −24.48878443668254534183448721450, −24.059049016261865302474167706244, −22.930227185269637044274377518535, −22.484612380241191163674160547355, −21.46906138584103958701660073385, −20.48583036673692301080922541404, −19.05100208955240178630716246584, −17.97006001228924235932052225855, −17.47822058481145521456062348248, −15.97495266979544986926165230498, −15.67731685848098348920548309705, −15.26047941728999893238061494115, −13.58257252736780657478833849742, −12.49333865746834212476901957787, −11.82133382732967445561675017237, −10.52953010439312391083810318654, −9.287973291533218662057076196647, −8.36360628245949300583924009003, −7.37481180247880272573974085111, −5.99985367357690285081607351915, −5.25455198312383771380977316921, −4.41168011352426808255137327878, −3.13329450877638096260571098801, −0.31152160128675972858583688414, 1.17914793236521577165250438635, 2.66857754850367049702575003009, 4.07000772540966903109645047283, 4.83200344740264595372161183357, 6.35140975757512994780843847766, 7.42554219009345457189067095526, 8.42898317591863705433448341250, 10.16782834975911694698779327376, 10.78575968654093898846917961676, 11.60511026537183536390267984782, 12.3811326538717216298771462314, 13.49491657006288064326811636063, 14.12655994161922391757227937128, 15.71952537868375318580412315368, 16.62648084435401501667840592647, 17.925564356622861851654267931403, 18.44733675314422994955105222734, 19.4272936087603468944043078561, 20.12981386614574804754901420978, 21.247342456187021720827004018178, 22.36074032647569490994837281016, 23.01296024480523103471898537609, 23.70739445206481273379380900723, 24.2880636865736364326301087117, 26.38194089672346188878746357048

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