Properties

Label 1-569-569.119-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.807 + 0.590i$
Analytic cond. $2.64242$
Root an. cond. $2.64242$
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  + (−0.110 + 0.993i)2-s + (−0.937 + 0.346i)3-s + (−0.975 − 0.219i)4-s + (−0.448 + 0.894i)5-s + (−0.240 − 0.970i)6-s + (−0.999 + 0.0442i)7-s + (0.325 − 0.945i)8-s + (0.759 − 0.650i)9-s + (−0.839 − 0.544i)10-s + (0.952 + 0.304i)11-s + (0.991 − 0.132i)12-s + (0.633 − 0.773i)13-s + (0.0663 − 0.997i)14-s + (0.110 − 0.993i)15-s + (0.903 + 0.428i)16-s + (−0.367 + 0.930i)17-s + ⋯
L(s)  = 1  + (−0.110 + 0.993i)2-s + (−0.937 + 0.346i)3-s + (−0.975 − 0.219i)4-s + (−0.448 + 0.894i)5-s + (−0.240 − 0.970i)6-s + (−0.999 + 0.0442i)7-s + (0.325 − 0.945i)8-s + (0.759 − 0.650i)9-s + (−0.839 − 0.544i)10-s + (0.952 + 0.304i)11-s + (0.991 − 0.132i)12-s + (0.633 − 0.773i)13-s + (0.0663 − 0.997i)14-s + (0.110 − 0.993i)15-s + (0.903 + 0.428i)16-s + (−0.367 + 0.930i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.807 + 0.590i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.807 + 0.590i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2198915364 + 0.6728860210i\)
\(L(\frac12)\) \(\approx\) \(0.2198915364 + 0.6728860210i\)
\(L(1)\) \(\approx\) \(0.4808516377 + 0.4474264752i\)
\(L(1)\) \(\approx\) \(0.4808516377 + 0.4474264752i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.110 + 0.993i)T \)
3 \( 1 + (-0.937 + 0.346i)T \)
5 \( 1 + (-0.448 + 0.894i)T \)
7 \( 1 + (-0.999 + 0.0442i)T \)
11 \( 1 + (0.952 + 0.304i)T \)
13 \( 1 + (0.633 - 0.773i)T \)
17 \( 1 + (-0.367 + 0.930i)T \)
19 \( 1 + (0.975 - 0.219i)T \)
23 \( 1 + (0.921 + 0.387i)T \)
29 \( 1 + (0.367 + 0.930i)T \)
31 \( 1 + (0.787 - 0.616i)T \)
37 \( 1 + (0.999 - 0.0442i)T \)
41 \( 1 + (-0.921 - 0.387i)T \)
43 \( 1 + (-0.110 - 0.993i)T \)
47 \( 1 + (-0.562 + 0.826i)T \)
53 \( 1 + (-0.240 - 0.970i)T \)
59 \( 1 + (-0.562 + 0.826i)T \)
61 \( 1 + (0.984 - 0.176i)T \)
67 \( 1 + (-0.367 - 0.930i)T \)
71 \( 1 + (0.325 + 0.945i)T \)
73 \( 1 + (0.975 - 0.219i)T \)
79 \( 1 + (-0.883 - 0.467i)T \)
83 \( 1 + (-0.996 + 0.0883i)T \)
89 \( 1 + (0.787 + 0.616i)T \)
97 \( 1 + (0.598 + 0.801i)T \)
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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

−22.99463107469637725366534622698, −22.1196392922616012183738008592, −21.30681204540299676067530875892, −20.327933052883599492672008846793, −19.53258601670064678610279079529, −18.90068146195790931348342788641, −18.098199940876736254583555029785, −16.95056165487681627154199479422, −16.53680210160917947707272328650, −15.68904448366382089918068572225, −13.92936367821700610274268587771, −13.29622172017137861854742148610, −12.4999892409870681689429043915, −11.62085544195906917496019436536, −11.384844538355980702977567879274, −9.94203626894308623064532659829, −9.29678658557632151094504151338, −8.35705867614011332437292747789, −7.05650879737339821731605737445, −6.07933298089111120623152310144, −4.88782914838480849174373627091, −4.158925045348467463988771700933, −3.04120295434713655243128460634, −1.43609958940823672285936208227, −0.650668756427403004154296005369, 0.96404445952259133580969729609, 3.32627990101452054221717056191, 3.96809537866082151289962207061, 5.21014322662563973204490027915, 6.29003194156636946269580815439, 6.636284805357080614164778757070, 7.57711401613388934120905878707, 8.893588686840517727069884302555, 9.82582031621712254192109519231, 10.531670630897425709055917937684, 11.54723804709838981554989571357, 12.60437757129719779016416386, 13.44233727568926863957541354424, 14.66721892257075158328505933590, 15.42331474380689589684109114386, 15.88247944920327131727916780696, 16.86504581112751547614603053598, 17.57746439880974416321292012308, 18.36602711113221881890105795807, 19.15062426568514788643998898644, 20.042445335137870010456338613470, 21.66150626149859624908387626801, 22.35046865493856461073084685937, 22.73968560885944668121216303463, 23.40329341916223423445410301614

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