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Properties

Label 1-17-17.3-r1-0-0
Degree $1$
Conductor $17$
Sign $0.825 - 0.563i$
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 + (0.923 + 0.382i)3-s − i·4-s + (−0.382 + 0.923i)5-s + (0.923 − 0.382i)6-s + (−0.382 − 0.923i)7-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (0.382 + 0.923i)10-s + (−0.923 + 0.382i)11-s + (0.382 − 0.923i)12-s + i·13-s + (−0.923 − 0.382i)14-s + (−0.707 + 0.707i)15-s − 16-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (0.923 + 0.382i)3-s − i·4-s + (−0.382 + 0.923i)5-s + (0.923 − 0.382i)6-s + (−0.382 − 0.923i)7-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (0.382 + 0.923i)10-s + (−0.923 + 0.382i)11-s + (0.382 − 0.923i)12-s + i·13-s + (−0.923 − 0.382i)14-s + (−0.707 + 0.707i)15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.825 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.825 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(17\)
Sign: $0.825 - 0.563i$
Analytic conductor: \(1.82690\)
Root analytic conductor: \(1.82690\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 17,\ (1:\ ),\ 0.825 - 0.563i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.840176969 - 0.5683876217i\)
\(L(\frac12)\) \(\approx\) \(1.840176969 - 0.5683876217i\)
\(L(1)\) \(\approx\) \(1.601018356 - 0.3927743947i\)
\(L(1)\) \(\approx\) \(1.601018356 - 0.3927743947i\)

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 \( 1 + (0.923 + 0.382i)T \)
5 \( 1 + (-0.382 + 0.923i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 + iT \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + (-0.923 - 0.382i)T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (-0.382 - 0.923i)T \)
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 \( 1 + (0.382 + 0.923i)T \)
67 \( 1 - T \)
71 \( 1 + (0.923 + 0.382i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.382 - 0.923i)T \)
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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

−41.65323987285134220259858973271, −40.27339812826001888244580245722, −38.91874155992885368810518458846, −37.13295793077064265061505583342, −35.64013803892604917260216774223, −34.780228724629630593423218653551, −32.72539385669846024139506052155, −31.71635953614421929616383063506, −31.03356287429880353240626572945, −29.19819432418985816304961338771, −27.09526202085607845668823357721, −25.4534845382737813313063178993, −24.67536749472359774974969489193, −23.37256169213771993885544259767, −21.4457288751630009563315276924, −20.16805365293293228901629694862, −18.29658711683568404583581042989, −16.1370625505243793736745381069, −15.08948043180965154777665971151, −13.25926508920288807793805024749, −12.36356835450350969236681067984, −8.89291284661477505556827442935, −7.71972133544652081438822773681, −5.432051278064953406097622403, −3.16711075027993145263853882, 2.76709448198816416655662751888, 4.27218557517400748548055345135, 7.162390560226882352945119354540, 9.75854802093811489627795191217, 11.02131628444003477640997179942, 13.26577436173584387516768142941, 14.42418557286325225650823050816, 15.74841843590598773856371535688, 18.71331050288806291606370812484, 19.82757322441046247619632825665, 21.08612712929105260798887509290, 22.54222792750195125378914065321, 23.86813184871099079942706211527, 25.99112907078403410387067861606, 27.03524747256711623408936707860, 28.96846965737148135531620437605, 30.49334852836046047606501340170, 31.21204385185497133225527849782, 32.6779217949817317264432997110, 33.781296017591045360792929367389, 36.19564635020667518488833083851, 37.33290540260283670869258826027, 38.71030887733946944097871582558, 39.18406921738597757547237818760, 41.356594899673041909870744826531

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