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Properties

Label 1-269-269.26-r1-0-0
Degree $1$
Conductor $269$
Sign $0.879 + 0.475i$
Analytic cond. $28.9080$
Root an. cond. $28.9080$
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  + (−0.977 − 0.209i)2-s + (0.533 + 0.845i)3-s + (0.912 + 0.409i)4-s + (0.995 − 0.0936i)5-s + (−0.344 − 0.938i)6-s + (0.646 + 0.762i)7-s + (−0.806 − 0.591i)8-s + (−0.430 + 0.902i)9-s + (−0.993 − 0.116i)10-s + (−0.163 − 0.986i)11-s + (0.140 + 0.990i)12-s + (0.116 − 0.993i)13-s + (−0.472 − 0.881i)14-s + (0.610 + 0.792i)15-s + (0.664 + 0.747i)16-s + (0.986 + 0.163i)17-s + ⋯
L(s)  = 1  + (−0.977 − 0.209i)2-s + (0.533 + 0.845i)3-s + (0.912 + 0.409i)4-s + (0.995 − 0.0936i)5-s + (−0.344 − 0.938i)6-s + (0.646 + 0.762i)7-s + (−0.806 − 0.591i)8-s + (−0.430 + 0.902i)9-s + (−0.993 − 0.116i)10-s + (−0.163 − 0.986i)11-s + (0.140 + 0.990i)12-s + (0.116 − 0.993i)13-s + (−0.472 − 0.881i)14-s + (0.610 + 0.792i)15-s + (0.664 + 0.747i)16-s + (0.986 + 0.163i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(269\)
Sign: $0.879 + 0.475i$
Analytic conductor: \(28.9080\)
Root analytic conductor: \(28.9080\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{269} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 269,\ (1:\ ),\ 0.879 + 0.475i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.068159810 + 0.5235683573i\)
\(L(\frac12)\) \(\approx\) \(2.068159810 + 0.5235683573i\)
\(L(1)\) \(\approx\) \(1.166841436 + 0.2047850817i\)
\(L(1)\) \(\approx\) \(1.166841436 + 0.2047850817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 \)
good2 \( 1 + (-0.977 - 0.209i)T \)
3 \( 1 + (0.533 + 0.845i)T \)
5 \( 1 + (0.995 - 0.0936i)T \)
7 \( 1 + (0.646 + 0.762i)T \)
11 \( 1 + (-0.163 - 0.986i)T \)
13 \( 1 + (0.116 - 0.993i)T \)
17 \( 1 + (0.986 + 0.163i)T \)
19 \( 1 + (0.997 - 0.0702i)T \)
23 \( 1 + (0.845 - 0.533i)T \)
29 \( 1 + (0.186 - 0.982i)T \)
31 \( 1 + (0.322 + 0.946i)T \)
37 \( 1 + (0.731 - 0.681i)T \)
41 \( 1 + (-0.209 - 0.977i)T \)
43 \( 1 + (-0.982 - 0.186i)T \)
47 \( 1 + (0.892 + 0.451i)T \)
53 \( 1 + (-0.869 - 0.493i)T \)
59 \( 1 + (-0.409 + 0.912i)T \)
61 \( 1 + (-0.819 + 0.572i)T \)
67 \( 1 + (-0.912 + 0.409i)T \)
71 \( 1 + (0.881 + 0.472i)T \)
73 \( 1 + (0.553 - 0.833i)T \)
79 \( 1 + (0.998 + 0.0468i)T \)
83 \( 1 + (-0.777 + 0.628i)T \)
89 \( 1 + (0.698 - 0.715i)T \)
97 \( 1 + (0.819 + 0.572i)T \)
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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

−25.48885299246616564645659595713, −24.870414903006252228581709409355, −23.883636374430474847741418035911, −23.19702990328561960326350951624, −21.40739345437110356240252580191, −20.56120554227146493902398658378, −20.02251048682835378168030871158, −18.67755806032525463212576566363, −18.23404455671606924645301235373, −17.28538621319760132695428184832, −16.684800062053936577680254396052, −15.0561354160522793938515248859, −14.289082021951308738421622646411, −13.50560636010397166060914359586, −12.17613976691825719228848369968, −11.15345627885068545249055615659, −9.85087282955847284475687584585, −9.309187573502313729403461844661, −7.98624467131292286076121779128, −7.237945446630433821726034479528, −6.43680955321896162722337626274, −5.07608578191354969524444490891, −3.04179935871461185650749730595, −1.751448510365896235081630133021, −1.16837106958815847816823931539, 1.07726192072235606680608119592, 2.51371256449622489885213836261, 3.23936065041225923520542106067, 5.20304642186200288649777217284, 5.94753687074475843801901334753, 7.75887440699455096320407964523, 8.57231812190185205896689078817, 9.313789616931390595043207231331, 10.29120301719465033679485658455, 10.975691954572282586828079226266, 12.217101467243577003610849539746, 13.55841139041200993427418515109, 14.60824411755157181826358413162, 15.54680895713043122461125957617, 16.44898160941603752691122827690, 17.34137267274324980122127509856, 18.2790984160953936871459817330, 19.09413403285368215189971092097, 20.28419304185877271104775001603, 21.09631485704719054514419874622, 21.45280464312905118377371487976, 22.474950969563348435710204216205, 24.3552037310270738477281780560, 25.09790408647506858422160843021, 25.54293728373360127933114974674

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