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Properties

Label 1-1049-1049.1009-r0-0-0
Degree $1$
Conductor $1049$
Sign $0.999 + 0.0238i$
Analytic cond. $4.87153$
Root an. cond. $4.87153$
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  + (−0.897 − 0.440i)2-s + (−0.306 + 0.951i)3-s + (0.612 + 0.790i)4-s + (−0.649 + 0.760i)5-s + (0.694 − 0.719i)6-s + (−0.818 + 0.574i)7-s + (−0.202 − 0.979i)8-s + (−0.811 − 0.583i)9-s + (0.918 − 0.396i)10-s + (0.0599 + 0.998i)11-s + (−0.940 + 0.340i)12-s + (0.944 − 0.329i)13-s + (0.987 − 0.155i)14-s + (−0.524 − 0.851i)15-s + (−0.249 + 0.968i)16-s + (0.374 − 0.927i)17-s + ⋯
L(s)  = 1  + (−0.897 − 0.440i)2-s + (−0.306 + 0.951i)3-s + (0.612 + 0.790i)4-s + (−0.649 + 0.760i)5-s + (0.694 − 0.719i)6-s + (−0.818 + 0.574i)7-s + (−0.202 − 0.979i)8-s + (−0.811 − 0.583i)9-s + (0.918 − 0.396i)10-s + (0.0599 + 0.998i)11-s + (−0.940 + 0.340i)12-s + (0.944 − 0.329i)13-s + (0.987 − 0.155i)14-s + (−0.524 − 0.851i)15-s + (−0.249 + 0.968i)16-s + (0.374 − 0.927i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $0.999 + 0.0238i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ 0.999 + 0.0238i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4938671832 + 0.005879072565i\)
\(L(\frac12)\) \(\approx\) \(0.4938671832 + 0.005879072565i\)
\(L(1)\) \(\approx\) \(0.5021861447 + 0.1197393303i\)
\(L(1)\) \(\approx\) \(0.5021861447 + 0.1197393303i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (-0.897 - 0.440i)T \)
3 \( 1 + (-0.306 + 0.951i)T \)
5 \( 1 + (-0.649 + 0.760i)T \)
7 \( 1 + (-0.818 + 0.574i)T \)
11 \( 1 + (0.0599 + 0.998i)T \)
13 \( 1 + (0.944 - 0.329i)T \)
17 \( 1 + (0.374 - 0.927i)T \)
19 \( 1 + (-0.965 - 0.260i)T \)
23 \( 1 + (0.996 + 0.0838i)T \)
29 \( 1 + (-0.736 - 0.676i)T \)
31 \( 1 + (0.991 - 0.131i)T \)
37 \( 1 + (-0.804 + 0.593i)T \)
41 \( 1 + (-0.260 + 0.965i)T \)
43 \( 1 + (0.272 - 0.962i)T \)
47 \( 1 + (0.583 - 0.811i)T \)
53 \( 1 + (-0.107 - 0.994i)T \)
59 \( 1 + (0.155 - 0.987i)T \)
61 \( 1 + (-0.667 - 0.744i)T \)
67 \( 1 + (-0.167 - 0.985i)T \)
71 \( 1 + (-0.694 - 0.719i)T \)
73 \( 1 + (0.887 - 0.461i)T \)
79 \( 1 + (0.667 - 0.744i)T \)
83 \( 1 + (-0.237 + 0.971i)T \)
89 \( 1 + (0.818 + 0.574i)T \)
97 \( 1 + (0.908 - 0.418i)T \)
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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

−21.28419820355231377738016174098, −20.47882988560885256174486406696, −19.521713123379162663643424832479, −19.11654953150747799383393151858, −18.71560878750419761808049235721, −17.4091193084406824981798587717, −16.88136052828692972198029393215, −16.35349920233567064673049196401, −15.63026513408568372956707716830, −14.478513423475001853850153414991, −13.53495561696588747489904553345, −12.82587303613054877541638590172, −11.98402638761658615525269037370, −10.957175444016816899344925055886, −10.58178416681386825523097703063, −9.017649147087516934034236988749, −8.633499423512691365359723625583, −7.79611727449717856924498849312, −6.97498530852916435240672002692, −6.17172666815771549967556076518, −5.56151912355999996795795671438, −4.10632869753178512554430285376, −2.98098787652902015839950249746, −1.46278029707966883316664410975, −0.85266698427596363104112726243, 0.41584350541002514730946253749, 2.25799900013367830211413550807, 3.16998442064684174411219297141, 3.73659250318041301022375142793, 4.893500692700115809792069449157, 6.28610472465528561650080044593, 6.83004019783024850718778932716, 7.96485407147784621594870886835, 8.87645510636839789440345985021, 9.61239355552641119769447467304, 10.30329018365190195923758337275, 11.01169502049751860157521338939, 11.770221393307061513003329370757, 12.39847531636990095905176458358, 13.51937413229101542749023700511, 15.14990772912626947324755752136, 15.25401555051353243932787974438, 16.04276552362786006114030829789, 16.87215157249029594650026208853, 17.67106394165286156174470397907, 18.55732066387092527518624390132, 19.08193435960834792274375039867, 19.982646016388158512614739236812, 20.70964835726904206625297950616, 21.34829438678476608140528289805

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