Properties

Label 1-812-812.51-r1-0-0
Degree 11
Conductor 812812
Sign 0.7990.601i-0.799 - 0.601i
Analytic cond. 87.261587.2615
Root an. cond. 87.261587.2615
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  + (−0.988 + 0.149i)3-s + (0.826 + 0.563i)5-s + (0.955 − 0.294i)9-s + (0.955 + 0.294i)11-s + (−0.222 − 0.974i)13-s + (−0.900 − 0.433i)15-s + (0.5 − 0.866i)17-s + (−0.988 − 0.149i)19-s + (−0.0747 − 0.997i)23-s + (0.365 + 0.930i)25-s + (−0.900 + 0.433i)27-s + (0.0747 − 0.997i)31-s + (−0.988 − 0.149i)33-s + (−0.955 + 0.294i)37-s + (0.365 + 0.930i)39-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)3-s + (0.826 + 0.563i)5-s + (0.955 − 0.294i)9-s + (0.955 + 0.294i)11-s + (−0.222 − 0.974i)13-s + (−0.900 − 0.433i)15-s + (0.5 − 0.866i)17-s + (−0.988 − 0.149i)19-s + (−0.0747 − 0.997i)23-s + (0.365 + 0.930i)25-s + (−0.900 + 0.433i)27-s + (0.0747 − 0.997i)31-s + (−0.988 − 0.149i)33-s + (−0.955 + 0.294i)37-s + (0.365 + 0.930i)39-s + ⋯

Functional equation

Λ(s)=(812s/2ΓR(s+1)L(s)=((0.7990.601i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(812s/2ΓR(s+1)L(s)=((0.7990.601i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 812812    =    227292^{2} \cdot 7 \cdot 29
Sign: 0.7990.601i-0.799 - 0.601i
Analytic conductor: 87.261587.2615
Root analytic conductor: 87.261587.2615
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ812(51,)\chi_{812} (51, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 812, (1: ), 0.7990.601i)(1,\ 812,\ (1:\ ),\ -0.799 - 0.601i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.15583543820.4664121535i0.1558354382 - 0.4664121535i
L(12)L(\frac12) \approx 0.15583543820.4664121535i0.1558354382 - 0.4664121535i
L(1)L(1) \approx 0.8198278747+0.006356008739i0.8198278747 + 0.006356008739i
L(1)L(1) \approx 0.8198278747+0.006356008739i0.8198278747 + 0.006356008739i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
7 1 1
29 1 1
good3 1+(0.988+0.149i)T 1 + (-0.988 + 0.149i)T
5 1+(0.826+0.563i)T 1 + (0.826 + 0.563i)T
11 1+(0.955+0.294i)T 1 + (0.955 + 0.294i)T
13 1+(0.2220.974i)T 1 + (-0.222 - 0.974i)T
17 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
19 1+(0.9880.149i)T 1 + (-0.988 - 0.149i)T
23 1+(0.07470.997i)T 1 + (-0.0747 - 0.997i)T
31 1+(0.07470.997i)T 1 + (0.0747 - 0.997i)T
37 1+(0.955+0.294i)T 1 + (-0.955 + 0.294i)T
41 1T 1 - T
43 1+(0.9000.433i)T 1 + (-0.900 - 0.433i)T
47 1+(0.733+0.680i)T 1 + (-0.733 + 0.680i)T
53 1+(0.07470.997i)T 1 + (0.0747 - 0.997i)T
59 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
61 1+(0.365+0.930i)T 1 + (-0.365 + 0.930i)T
67 1+(0.733+0.680i)T 1 + (0.733 + 0.680i)T
71 1+(0.222+0.974i)T 1 + (0.222 + 0.974i)T
73 1+(0.826+0.563i)T 1 + (-0.826 + 0.563i)T
79 1+(0.9550.294i)T 1 + (0.955 - 0.294i)T
83 1+(0.623+0.781i)T 1 + (-0.623 + 0.781i)T
89 1+(0.8260.563i)T 1 + (-0.826 - 0.563i)T
97 1+(0.623+0.781i)T 1 + (-0.623 + 0.781i)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

−22.17003634714836981200357139040, −21.48014621167595615583934453864, −21.22997827501073372155176588340, −19.777998934676991018769763618624, −19.135204707662800558886634274723, −18.19077222452700048996399827904, −17.24637427330324944494767266127, −16.92686016084669263967618190163, −16.25929960036928622217465387601, −15.08599144558639533452532687721, −14.07576214730531707913935891311, −13.34446217145436382810318818178, −12.36480729692755199088784586108, −11.881759838475161010245552323486, −10.82646987501170492945985208513, −10.01964134706044673174417871590, −9.1856190649771899902612262791, −8.2838083419890127366036103012, −6.88459902872294734866747365662, −6.333961176762971478148541521568, −5.47735961924563113061741085629, −4.61548693231450455146435867545, −3.64327486201707117996173665720, −1.802272846948377334567210212025, −1.39235333905559225638131489973, 0.12386831344812600006931317190, 1.34019451265072274498330152508, 2.48874868856449713808025622299, 3.709185590903733997070110628711, 4.85026725113815546147308585804, 5.61847178968723910083117385503, 6.55893712107154699227859931734, 7.03112057744706380218394748006, 8.40195906656099230800261299733, 9.664513343979317211675749034471, 10.104503199942107532435539266829, 10.96968497020187775938022403890, 11.81100631782120833921036960480, 12.65315835292740419183603217750, 13.46488624227166270672254897233, 14.57807763902686406125786704716, 15.124576348470050106359188015881, 16.24890195815007706158326536, 17.12955994987080210534129105744, 17.492039065493688996378479276799, 18.40895628519290153512382567798, 19.030539232013679840938683140, 20.31149198542435615593778966465, 21.02236637884526625189396960534, 21.954955478252055628889348243120

Graph of the ZZ-function along the critical line