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Properties

Label 1-17-17.16-r0-0-0
Degree $1$
Conductor $17$
Sign $1$
Analytic cond. $0.0789476$
Root an. cond. $0.0789476$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 11-s − 12-s + 13-s − 14-s + 15-s + 16-s + 18-s + 19-s − 20-s + 21-s − 22-s − 23-s − 24-s + 25-s + 26-s − 27-s − 28-s − 29-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 11-s − 12-s + 13-s − 14-s + 15-s + 16-s + 18-s + 19-s − 20-s + 21-s − 22-s − 23-s − 24-s + 25-s + 26-s − 27-s − 28-s − 29-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(17\)
Sign: $1$
Analytic conductor: \(0.0789476\)
Root analytic conductor: \(0.0789476\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{17} (16, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 17,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7216743742\)
\(L(\frac12)\) \(\approx\) \(0.7216743742\)
\(L(1)\) \(\approx\) \(1.016084833\)
\(L(1)\) \(\approx\) \(1.016084833\)

Euler product

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

−41.54780485772929269257760065165, −40.02360201806485561135697055902, −39.16823527643206989170389643189, −38.27607125410327060500653430616, −35.59321847652579937308587748261, −34.649603957105146560939758493718, −33.33544614659282345208173853118, −32.00478870264517330192920665826, −30.6688830316828123033048961035, −29.19674226270232323296465712101, −28.13750838550913232632989000419, −26.107272098738978805398854175757, −24.080673662839181440958833964353, −23.15205504769474128245193123053, −22.200028093892264290603734681987, −20.42753799212527225307532563642, −18.681633236191055643020999527797, −16.26848320767812805605363828402, −15.643514532468074372287370378015, −13.20980729951037452362304975494, −11.97767299241460230215255443712, −10.61732009264966691128038885347, −7.28283285953425231462503990159, −5.6355907582091547668150421442, −3.72814208504204479132367863258, 3.72814208504204479132367863258, 5.6355907582091547668150421442, 7.28283285953425231462503990159, 10.61732009264966691128038885347, 11.97767299241460230215255443712, 13.20980729951037452362304975494, 15.643514532468074372287370378015, 16.26848320767812805605363828402, 18.681633236191055643020999527797, 20.42753799212527225307532563642, 22.200028093892264290603734681987, 23.15205504769474128245193123053, 24.080673662839181440958833964353, 26.107272098738978805398854175757, 28.13750838550913232632989000419, 29.19674226270232323296465712101, 30.6688830316828123033048961035, 32.00478870264517330192920665826, 33.33544614659282345208173853118, 34.649603957105146560939758493718, 35.59321847652579937308587748261, 38.27607125410327060500653430616, 39.16823527643206989170389643189, 40.02360201806485561135697055902, 41.54780485772929269257760065165

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