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Properties

Label 1-17-17.13-r0-0-0
Degree 11
Conductor 1717
Sign 0.615+0.788i0.615 + 0.788i
Analytic cond. 0.07894760.0789476
Root an. cond. 0.07894760.0789476
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(17s/2 ΓR(s) L(s)=((0.615+0.788i) Λ‾(1−s)\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}
Λ(s)=(17s/2 ΓR(s) L(s)=((0.615+0.788i) Λ‾(1−s)\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: 11
Conductor: 1717
Sign: 0.615+0.788i0.615 + 0.788i
Analytic conductor: 0.07894760.0789476
Root analytic conductor: 0.07894760.0789476
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ17(13,⋅)\chi_{17} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 17, (0: ), 0.615+0.788i)(1,\ 17,\ (0:\ ),\ 0.615 + 0.788i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.3883506224+0.1894872842i0.3883506224 + 0.1894872842i
L(12)L(\frac12) ≈\approx 0.3883506224+0.1894872842i0.3883506224 + 0.1894872842i
L(1)L(1) ≈\approx 0.5927817455+0.1936151527i0.5927817455 + 0.1936151527i
L(1)L(1) ≈\approx 0.5927817455+0.1936151527i0.5927817455 + 0.1936151527i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad17 1 1
good2 1 1
3 1+T 1 + T
5 1−T 1 - T
7 1+iT 1 + iT
11 1+T 1 + T
13 1+iT 1 + iT
19 1−iT 1 - iT
23 1−T 1 - T
29 1−T 1 - T
31 1−iT 1 - iT
37 1−iT 1 - iT
41 1+iT 1 + iT
43 1+T 1 + T
47 1+iT 1 + iT
53 1−T 1 - T
59 1+T 1 + T
61 1 1
67 1+T 1 + T
71 1−T 1 - T
73 1+iT 1 + iT
79 1+T 1 + T
83 1+iT 1 + iT
89 1−iT 1 - iT
97 1−iT 1 - iT
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   L(s)=∏p (1−αp p−s)−1L(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 ZZ-function along the critical line