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Properties

Label 1-17-17.13-r0-0-0
Degree $1$
Conductor $17$
Sign $0.615 + 0.788i$
Analytic cond. $0.0789476$
Root an. cond. $0.0789476$
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  − 2-s + i·3-s + 4-s + i·5-s − i·6-s − i·7-s − 8-s − 9-s − i·10-s − i·11-s + i·12-s + 13-s + i·14-s − 15-s + 16-s + ⋯
L(s)  = 1  − 2-s + i·3-s + 4-s + i·5-s − i·6-s − i·7-s − 8-s − 9-s − i·10-s − i·11-s + i·12-s + 13-s + i·14-s − 15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(17\)
Sign: $0.615 + 0.788i$
Analytic conductor: \(0.0789476\)
Root analytic conductor: \(0.0789476\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 17,\ (0:\ ),\ 0.615 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3883506224 + 0.1894872842i\)
\(L(\frac12)\) \(\approx\) \(0.3883506224 + 0.1894872842i\)
\(L(1)\) \(\approx\) \(0.5927817455 + 0.1936151527i\)
\(L(1)\) \(\approx\) \(0.5927817455 + 0.1936151527i\)

Euler product

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

−41.39825263368656834810158053634, −40.123878538062289210737176714428, −38.3670491737333797985499135149, −36.9750433058488300935484291589, −35.83300407184436397685818801208, −35.14296567168541579991627230428, −33.60034550680774720031512265464, −31.5696493152975137686651010417, −30.10578496434668135472201722168, −28.522996391203429046323481231197, −28.00152292947194639614103841511, −25.62481024876169433791299063204, −24.91955176224272177116750484121, −23.53276396731361401358774471776, −20.964356893460047508874269655244, −19.6158544746063732711902498863, −18.30609178276429653661407253897, −17.122130095672791600965426780671, −15.40203147631056025262030921185, −12.86274574324480776117856226684, −11.65632735313783772341642685163, −9.19390267662336140187995090804, −7.97395225057117766202200498016, −6.02106312276084192766295517421, −1.89456288883640063451868939480, 3.38764301980058083148824334443, 6.466503328866556453198478281136, 8.51273109083455322179148702773, 10.41283347696993241919453381820, 11.03897117769565301368588396339, 14.27914098300065745959487204470, 15.854658931004722037734299863, 17.09285647216109024420336241365, 18.76358864398668875311125644434, 20.28073895145710467217235457053, 21.6437758792531046000649108338, 23.41778677783755363216066933962, 25.69146946873353330579060148931, 26.59620981521950272346296366303, 27.49201032500673839848802587111, 29.14964593988215408206112800369, 30.445439668547355789741455455925, 32.676185030466947999125108619010, 33.76074984228369908790641957324, 34.81856497845666396290521055952, 36.601442066436625569348311303850, 37.78303357163219485745268114579, 38.64551375754757981318534880869, 39.95820595043727429615498631813, 42.537824722990093368896081175906

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