Properties

Label 1-20e2-400.27-r0-0-0
Degree 11
Conductor 400400
Sign 0.7800.625i-0.780 - 0.625i
Analytic cond. 1.857591.85759
Root an. cond. 1.857591.85759
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  + (−0.809 − 0.587i)3-s + i·7-s + (0.309 + 0.951i)9-s + (−0.951 − 0.309i)11-s + (−0.309 − 0.951i)13-s + (−0.587 − 0.809i)17-s + (0.587 + 0.809i)19-s + (0.587 − 0.809i)21-s + (−0.951 − 0.309i)23-s + (0.309 − 0.951i)27-s + (0.587 − 0.809i)29-s + (0.809 − 0.587i)31-s + (0.587 + 0.809i)33-s + (−0.309 − 0.951i)37-s + (−0.309 + 0.951i)39-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)3-s + i·7-s + (0.309 + 0.951i)9-s + (−0.951 − 0.309i)11-s + (−0.309 − 0.951i)13-s + (−0.587 − 0.809i)17-s + (0.587 + 0.809i)19-s + (0.587 − 0.809i)21-s + (−0.951 − 0.309i)23-s + (0.309 − 0.951i)27-s + (0.587 − 0.809i)29-s + (0.809 − 0.587i)31-s + (0.587 + 0.809i)33-s + (−0.309 − 0.951i)37-s + (−0.309 + 0.951i)39-s + ⋯

Functional equation

Λ(s)=(400s/2ΓR(s)L(s)=((0.7800.625i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(400s/2ΓR(s)L(s)=((0.7800.625i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 400400    =    24522^{4} \cdot 5^{2}
Sign: 0.7800.625i-0.780 - 0.625i
Analytic conductor: 1.857591.85759
Root analytic conductor: 1.857591.85759
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ400(27,)\chi_{400} (27, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 400, (0: ), 0.7800.625i)(1,\ 400,\ (0:\ ),\ -0.780 - 0.625i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.13613669010.3876605351i0.1361366901 - 0.3876605351i
L(12)L(\frac12) \approx 0.13613669010.3876605351i0.1361366901 - 0.3876605351i
L(1)L(1) \approx 0.61825848730.1588678174i0.6182584873 - 0.1588678174i
L(1)L(1) \approx 0.61825848730.1588678174i0.6182584873 - 0.1588678174i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
good3 1+(0.8090.587i)T 1 + (-0.809 - 0.587i)T
7 1+iT 1 + iT
11 1+(0.9510.309i)T 1 + (-0.951 - 0.309i)T
13 1+(0.3090.951i)T 1 + (-0.309 - 0.951i)T
17 1+(0.5870.809i)T 1 + (-0.587 - 0.809i)T
19 1+(0.587+0.809i)T 1 + (0.587 + 0.809i)T
23 1+(0.9510.309i)T 1 + (-0.951 - 0.309i)T
29 1+(0.5870.809i)T 1 + (0.587 - 0.809i)T
31 1+(0.8090.587i)T 1 + (0.809 - 0.587i)T
37 1+(0.3090.951i)T 1 + (-0.309 - 0.951i)T
41 1+(0.3090.951i)T 1 + (-0.309 - 0.951i)T
43 1T 1 - T
47 1+(0.587+0.809i)T 1 + (-0.587 + 0.809i)T
53 1+(0.8090.587i)T 1 + (-0.809 - 0.587i)T
59 1+(0.951+0.309i)T 1 + (-0.951 + 0.309i)T
61 1+(0.9510.309i)T 1 + (-0.951 - 0.309i)T
67 1+(0.8090.587i)T 1 + (0.809 - 0.587i)T
71 1+(0.8090.587i)T 1 + (-0.809 - 0.587i)T
73 1+(0.951+0.309i)T 1 + (0.951 + 0.309i)T
79 1+(0.8090.587i)T 1 + (-0.809 - 0.587i)T
83 1+(0.809+0.587i)T 1 + (-0.809 + 0.587i)T
89 1+(0.3090.951i)T 1 + (0.309 - 0.951i)T
97 1+(0.5870.809i)T 1 + (0.587 - 0.809i)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

−24.3720672533461280870528358469, −23.63542472818526135622546664668, −23.191831653544505384439653228531, −21.95362593893559641309110104648, −21.503569060429191687406317080926, −20.394163233916193663641125164545, −19.71576913514370114108269537847, −18.361142971987097318297789200312, −17.6003097099340392480132496349, −16.84315920656300787707281475505, −15.99529340806297218743183595387, −15.255237787850840062323023212681, −14.05965441997778211690833184554, −13.16458028899732984219976182003, −12.06621163534045202674342708488, −11.17978031956272659855497629827, −10.317745644295511603909274370014, −9.696352261582907299493026715903, −8.36050226928974737188661271015, −7.09638501580768999928069440205, −6.381547009304692897532133789520, −4.97063282181268719199447103415, −4.42379567512762485348245704886, −3.19423345920826780639846317626, −1.4849072559701040194802770840, 0.26498896849805662510875437267, 2.00087878739302388283604247619, 2.95435456911883077457593818685, 4.76405917149432609078017127825, 5.57927271547639402814615044344, 6.30163783896814579888458786961, 7.65639750771766760993762249074, 8.26463209896971486001690569609, 9.72372547786250085694270842604, 10.61817581394707898206445965017, 11.68282997268812460666525049617, 12.32467516278085997151267689551, 13.19121617801598011941641921787, 14.13026200053435489231345710019, 15.59593403640639191501174605065, 15.93763944177045399014237555274, 17.19366214537328049240175689356, 18.13477357573103748280546518222, 18.479752500674838801811055360067, 19.50849662342221793682060869695, 20.64638313306267980970407361878, 21.598128580299261312136888143840, 22.53783957252884157301762065807, 22.95807458352101776818197246393, 24.28246618916523510760160366728

Graph of the ZZ-function along the critical line