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Properties

Label 1-17-17.14-r1-0-0
Degree $1$
Conductor $17$
Sign $-0.139 - 0.990i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(17\)
Sign: $-0.139 - 0.990i$
Analytic conductor: \(1.82690\)
Root analytic conductor: \(1.82690\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 17,\ (1:\ ),\ -0.139 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9311494398 - 1.071252131i\)
\(L(\frac12)\) \(\approx\) \(0.9311494398 - 1.071252131i\)
\(L(1)\) \(\approx\) \(1.026557040 - 0.7114857552i\)
\(L(1)\) \(\approx\) \(1.026557040 - 0.7114857552i\)

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

−41.48946198471389814419910491101, −40.31591073775791236539514048537, −39.335895821080204099454304634676, −37.86254223053293509339853159624, −35.67345877705159363372997709813, −34.38300182012545449675241340593, −33.415988976217645313609009981041, −32.59142479550074655351488845417, −30.40688913720403841833202134516, −29.64646318212120248292668307050, −27.402890689538725399263918583460, −26.263220833794432448847595709004, −24.55088429721233190497002767122, −22.9160878133856775638251956415, −22.35274006712663527817276174439, −20.73601382461062255678490327801, −17.89162486314073795673833439090, −16.96255332088990513809409072366, −15.258727862890474415073123448188, −13.88726765877519019925207643961, −11.900561085775338517174447415529, −10.26525357790367285198891490709, −7.29713881290272697352366687991, −5.87613513547569449746020404318, −3.96733015716415479332555752191, 1.554269601485073702014516769817, 4.7768875978404875933929745300, 6.12454143257586315359542021887, 9.28558905646388458326732102136, 11.45953084658036273651653779807, 12.34958587404643874845085677239, 13.91997128725079869171238053916, 16.10500204401020727088348424817, 17.87907737409314949954753945566, 19.43553024609536487202111720969, 21.29930234154393680288336591490, 22.19566986157050046937089496603, 23.98511641809200456207271133468, 24.67870197679031250096714000512, 27.78030082620755135134436060800, 28.520825290047799633209856230773, 29.69276975087506697806333371884, 31.138367874699192222369633610588, 32.569625581196314344038725156453, 33.82607093372941793115291640616, 35.48248964918470882152905094983, 36.96290295731723323815929585104, 38.44645407061706632432161588107, 39.87164407935438441616508614389, 40.68959831521997260475436352804

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