Properties

Label 1-1009-1009.101-r0-0-0
Degree 11
Conductor 10091009
Sign 0.7910.610i-0.791 - 0.610i
Analytic cond. 4.685774.68577
Root an. cond. 4.685774.68577
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  + (0.124 + 0.992i)2-s + (0.733 + 0.680i)3-s + (−0.969 + 0.246i)4-s + (0.411 + 0.911i)5-s + (−0.583 + 0.811i)6-s + (0.542 − 0.840i)7-s + (−0.365 − 0.930i)8-s + (0.0747 + 0.997i)9-s + (−0.853 + 0.521i)10-s + (−0.911 + 0.411i)11-s + (−0.878 − 0.478i)12-s + (−0.866 + 0.5i)13-s + (0.900 + 0.433i)14-s + (−0.318 + 0.947i)15-s + (0.878 − 0.478i)16-s + (−0.478 − 0.878i)17-s + ⋯
L(s)  = 1  + (0.124 + 0.992i)2-s + (0.733 + 0.680i)3-s + (−0.969 + 0.246i)4-s + (0.411 + 0.911i)5-s + (−0.583 + 0.811i)6-s + (0.542 − 0.840i)7-s + (−0.365 − 0.930i)8-s + (0.0747 + 0.997i)9-s + (−0.853 + 0.521i)10-s + (−0.911 + 0.411i)11-s + (−0.878 − 0.478i)12-s + (−0.866 + 0.5i)13-s + (0.900 + 0.433i)14-s + (−0.318 + 0.947i)15-s + (0.878 − 0.478i)16-s + (−0.478 − 0.878i)17-s + ⋯

Functional equation

Λ(s)=(1009s/2ΓR(s)L(s)=((0.7910.610i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1009s/2ΓR(s)L(s)=((0.7910.610i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 10091009
Sign: 0.7910.610i-0.791 - 0.610i
Analytic conductor: 4.685774.68577
Root analytic conductor: 4.685774.68577
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1009(101,)\chi_{1009} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 1009, (0: ), 0.7910.610i)(1,\ 1009,\ (0:\ ),\ -0.791 - 0.610i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.4592565623+1.346937111i-0.4592565623 + 1.346937111i
L(12)L(\frac12) \approx 0.4592565623+1.346937111i-0.4592565623 + 1.346937111i
L(1)L(1) \approx 0.6579525861+1.004145273i0.6579525861 + 1.004145273i
L(1)L(1) \approx 0.6579525861+1.004145273i0.6579525861 + 1.004145273i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad1009 1 1
good2 1+(0.124+0.992i)T 1 + (0.124 + 0.992i)T
3 1+(0.733+0.680i)T 1 + (0.733 + 0.680i)T
5 1+(0.411+0.911i)T 1 + (0.411 + 0.911i)T
7 1+(0.5420.840i)T 1 + (0.542 - 0.840i)T
11 1+(0.911+0.411i)T 1 + (-0.911 + 0.411i)T
13 1+(0.866+0.5i)T 1 + (-0.866 + 0.5i)T
17 1+(0.4780.878i)T 1 + (-0.478 - 0.878i)T
19 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
23 1+(0.149+0.988i)T 1 + (0.149 + 0.988i)T
29 1+(0.456+0.889i)T 1 + (-0.456 + 0.889i)T
31 1+(0.9920.124i)T 1 + (-0.992 - 0.124i)T
37 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
41 1+(0.9390.342i)T 1 + (-0.939 - 0.342i)T
43 1+(0.866+0.5i)T 1 + (-0.866 + 0.5i)T
47 1+(0.2940.955i)T 1 + (-0.294 - 0.955i)T
53 1+(0.603+0.797i)T 1 + (0.603 + 0.797i)T
59 1+(0.4330.900i)T 1 + (-0.433 - 0.900i)T
61 1+(0.6800.733i)T 1 + (-0.680 - 0.733i)T
67 1+(0.542+0.840i)T 1 + (0.542 + 0.840i)T
71 1+(0.124+0.992i)T 1 + (0.124 + 0.992i)T
73 1+(0.563+0.826i)T 1 + (-0.563 + 0.826i)T
79 1+(0.947+0.318i)T 1 + (0.947 + 0.318i)T
83 1+(0.5210.853i)T 1 + (0.521 - 0.853i)T
89 1+(0.811+0.583i)T 1 + (0.811 + 0.583i)T
97 1+(0.715+0.698i)T 1 + (0.715 + 0.698i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−21.08371614621401008896364390600, −20.30817642993908111491920276072, −19.807333249695962147079581743893, −18.90949647223242553294743684579, −18.11441366542944200016643540478, −17.73514533263546463856212034890, −16.64389413889616910448656322994, −15.24388713136090212179487980493, −14.76045426227905854590108614927, −13.64507742521945761707775596803, −13.14152334737967011410714243596, −12.46981080611935852504070578883, −11.86092733242524632131262141422, −10.78497495343239361233371871751, −9.726194942853991769386765067341, −9.036475752487302519061316783539, −8.33059207696030290650893326799, −7.73513137583817518833800825375, −6.07465254832670418606744804000, −5.28428835102419505728507337697, −4.50358284808896982488116149000, −3.17337309738644207280616185816, −2.34553342338384439220382849341, −1.74797065676135816017784595100, −0.49226702022503345821614549239, 1.84937398104984311600250929908, 3.02735058084224014542440513099, 3.83048208831743930330563146021, 4.930396806708529792226501350875, 5.36293266112586503875272191408, 6.94913140535766580837439577656, 7.377766157912451291300016146424, 8.04373148218290487409146808633, 9.34355401375698315256639908851, 9.80062256090819925556648111878, 10.60265583185428644625938980354, 11.609567407171027786490028832117, 13.165185808864485226579098856839, 13.6733802557288772785677564009, 14.41572469555113284628260184385, 14.87461575902392429717913298110, 15.68601995687094369625140843215, 16.519143734619400575935413652462, 17.255108993893578778045862771001, 18.2173792057222369074268760197, 18.650248795675073748432092757052, 19.93340810779610483489131438821, 20.54921231125010337941899387231, 21.72608091014272193274275701465, 21.883944535439646439726094828784

Graph of the ZZ-function along the critical line