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Properties

Label 1-269-269.138-r0-0-0
Degree $1$
Conductor $269$
Sign $-0.522 + 0.852i$
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  + (0.912 + 0.409i)2-s + (−0.430 + 0.902i)3-s + (0.664 + 0.747i)4-s + (0.982 − 0.186i)5-s + (−0.762 + 0.646i)6-s + (−0.163 + 0.986i)7-s + (0.300 + 0.953i)8-s + (−0.628 − 0.777i)9-s + (0.972 + 0.232i)10-s + (−0.946 + 0.322i)11-s + (−0.960 + 0.277i)12-s + (−0.972 − 0.232i)13-s + (−0.553 + 0.833i)14-s + (−0.255 + 0.966i)15-s + (−0.116 + 0.993i)16-s + (0.946 + 0.322i)17-s + ⋯
L(s)  = 1  + (0.912 + 0.409i)2-s + (−0.430 + 0.902i)3-s + (0.664 + 0.747i)4-s + (0.982 − 0.186i)5-s + (−0.762 + 0.646i)6-s + (−0.163 + 0.986i)7-s + (0.300 + 0.953i)8-s + (−0.628 − 0.777i)9-s + (0.972 + 0.232i)10-s + (−0.946 + 0.322i)11-s + (−0.960 + 0.277i)12-s + (−0.972 − 0.232i)13-s + (−0.553 + 0.833i)14-s + (−0.255 + 0.966i)15-s + (−0.116 + 0.993i)16-s + (0.946 + 0.322i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(269\)
Sign: $-0.522 + 0.852i$
Analytic conductor: \(1.24923\)
Root analytic conductor: \(1.24923\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{269} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 269,\ (0:\ ),\ -0.522 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9499191151 + 1.695655494i\)
\(L(\frac12)\) \(\approx\) \(0.9499191151 + 1.695655494i\)
\(L(1)\) \(\approx\) \(1.282115032 + 1.018826026i\)
\(L(1)\) \(\approx\) \(1.282115032 + 1.018826026i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 \)
good2 \( 1 + (0.912 + 0.409i)T \)
3 \( 1 + (-0.430 + 0.902i)T \)
5 \( 1 + (0.982 - 0.186i)T \)
7 \( 1 + (-0.163 + 0.986i)T \)
11 \( 1 + (-0.946 + 0.322i)T \)
13 \( 1 + (-0.972 - 0.232i)T \)
17 \( 1 + (0.946 + 0.322i)T \)
19 \( 1 + (0.990 - 0.140i)T \)
23 \( 1 + (0.430 - 0.902i)T \)
29 \( 1 + (-0.930 - 0.366i)T \)
31 \( 1 + (-0.792 + 0.610i)T \)
37 \( 1 + (0.0702 - 0.997i)T \)
41 \( 1 + (-0.912 + 0.409i)T \)
43 \( 1 + (0.930 + 0.366i)T \)
47 \( 1 + (0.591 + 0.806i)T \)
53 \( 1 + (0.513 + 0.858i)T \)
59 \( 1 + (-0.664 - 0.747i)T \)
61 \( 1 + (0.344 - 0.938i)T \)
67 \( 1 + (0.664 - 0.747i)T \)
71 \( 1 + (0.553 + 0.833i)T \)
73 \( 1 + (-0.388 - 0.921i)T \)
79 \( 1 + (0.995 + 0.0936i)T \)
83 \( 1 + (0.209 - 0.977i)T \)
89 \( 1 + (-0.0234 - 0.999i)T \)
97 \( 1 + (0.344 + 0.938i)T \)
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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

−25.2495756184049732300963313775, −24.189753952478673426790156002013, −23.71394557109879248017865681955, −22.68117272855738798963450566393, −22.038118213317337429138236885024, −20.93445935744064103761298411802, −20.13962091520060665122937461795, −19.020518869618475822666779161885, −18.29686564598423006390766152386, −17.06462727963415160360068533570, −16.39147490678901095401506007635, −14.80111996479573654505245004564, −13.76621047840113660907920138227, −13.44358644444252254669288689089, −12.46685932789550095497960855629, −11.408686869409408282711689367374, −10.4515903332640262948099314760, −9.652527497294128327403400947, −7.51341597059192850989599569264, −6.99845290625400236868305640594, −5.61468196509053665265226327032, −5.183871533977615501201876634647, −3.357788548270417245295888987380, −2.27154964558707562142099191894, −1.08751870479458982049399333428, 2.30900037012692884308597747827, 3.22981556356350304837525272311, 4.89022227371268990302701652710, 5.36687796034836479673832993984, 6.14164931409451993589058671731, 7.585990665074592069526994460, 9.004117348448925328944499535721, 9.93498630551973137113472432799, 10.97565825832079065722566323823, 12.30436590508331397651250259004, 12.75197365762527349423649361439, 14.20586649123004164450092487397, 14.90488192897150078684924740496, 15.79560292500051246322065177791, 16.63365207537198580241187292482, 17.469324540087230279412211413530, 18.44921366783219621813297302253, 20.227420820201651676652441500333, 20.97745570512639367673195987622, 21.69750735771223530253863985733, 22.27264302457114570253215120111, 23.127953993175141347392230460733, 24.30246400554054474496813452590, 25.09707290571398426167570232393, 25.935373901145167142416968539814

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