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Properties

Label 1-17-17.5-r1-0-0
Degree $1$
Conductor $17$
Sign $-0.974 - 0.226i$
Analytic cond. $1.82690$
Root an. cond. $1.82690$
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  + (−0.707 + 0.707i)2-s + (−0.382 + 0.923i)3-s − i·4-s + (−0.923 − 0.382i)5-s + (−0.382 − 0.923i)6-s + (−0.923 + 0.382i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)10-s + (0.382 + 0.923i)11-s + (0.923 + 0.382i)12-s + i·13-s + (0.382 − 0.923i)14-s + (0.707 − 0.707i)15-s − 16-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.382 + 0.923i)3-s − i·4-s + (−0.923 − 0.382i)5-s + (−0.382 − 0.923i)6-s + (−0.923 + 0.382i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)10-s + (0.382 + 0.923i)11-s + (0.923 + 0.382i)12-s + i·13-s + (0.382 − 0.923i)14-s + (0.707 − 0.707i)15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.974 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.974 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(17\)
Sign: $-0.974 - 0.226i$
Analytic conductor: \(1.82690\)
Root analytic conductor: \(1.82690\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 17,\ (1:\ ),\ -0.974 - 0.226i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03785959933 + 0.3304265109i\)
\(L(\frac12)\) \(\approx\) \(-0.03785959933 + 0.3304265109i\)
\(L(1)\) \(\approx\) \(0.3364536872 + 0.3041433867i\)
\(L(1)\) \(\approx\) \(0.3364536872 + 0.3041433867i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 + (-0.923 - 0.382i)T \)
7 \( 1 + (-0.923 + 0.382i)T \)
11 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 + iT \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (-0.382 - 0.923i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 + (0.382 - 0.923i)T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (-0.923 + 0.382i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (0.923 - 0.382i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.382 + 0.923i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.923 + 0.382i)T \)
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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

−40.08933506006515232808569501134, −39.211164042317428677088923901181, −37.79132475493121548469594362424, −36.250160631826820748051406908228, −35.203900470717652026601521520725, −34.49681110980341751107496182124, −31.88421685697285491575892731058, −30.30074573595674422592446311993, −29.59897888664145406627834667602, −28.13248118072899166640715101029, −26.77719057098203746301071286072, −25.27940750172787845585789905595, −23.38375312098535499146615790182, −22.14392892474512541309571125350, −19.73391427861055133875059216657, −19.19445133339640635093713708536, −17.64120167489276756286468707560, −16.12148499173592123748792240253, −13.333629792910303461114000633001, −11.95784778691279866741936781380, −10.63861745698672798356651503622, −8.29867536775569339857252816986, −6.8194345706205735510214385089, −3.232520385351869734905576312524, −0.39131461302555943968992017527, 4.45150760621423341536424083440, 6.521377799321187945309143225667, 8.72077279114639747851188082385, 10.05473234794703825384217167606, 11.95487896113766360394246049370, 14.84726188590983910559904346551, 15.98742108078147625047328159174, 16.93645045266218797071450118487, 18.947383980442873732651471005600, 20.32897469362333379148347235501, 22.51119148271240252445779939656, 23.63587325283735955260546353974, 25.4886707480401936511815226321, 26.7494894689177514473207280004, 27.946773560061612964642362317315, 28.78234775346773357309072737393, 31.54570671160629637629440255225, 32.60751041348255207926129951700, 33.914463546227765267034293948626, 35.142633592076596907751364156085, 36.166647137494799468204170725424, 38.22430974545368939352183426367, 38.76340003441337217120280084163, 40.6851019417213447809555920791, 42.35923171085841664792465386996

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