Properties

Label 1-392-392.285-r1-0-0
Degree 11
Conductor 392392
Sign 0.04270.999i-0.0427 - 0.999i
Analytic cond. 42.126242.1262
Root an. cond. 42.126242.1262
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  + (0.955 + 0.294i)3-s + (−0.733 + 0.680i)5-s + (0.826 + 0.563i)9-s + (−0.826 + 0.563i)11-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)15-s + (−0.365 − 0.930i)17-s + (−0.5 − 0.866i)19-s + (0.365 − 0.930i)23-s + (0.0747 − 0.997i)25-s + (0.623 + 0.781i)27-s + (−0.623 + 0.781i)29-s + (0.5 − 0.866i)31-s + (−0.955 + 0.294i)33-s + (0.988 + 0.149i)37-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)3-s + (−0.733 + 0.680i)5-s + (0.826 + 0.563i)9-s + (−0.826 + 0.563i)11-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)15-s + (−0.365 − 0.930i)17-s + (−0.5 − 0.866i)19-s + (0.365 − 0.930i)23-s + (0.0747 − 0.997i)25-s + (0.623 + 0.781i)27-s + (−0.623 + 0.781i)29-s + (0.5 − 0.866i)31-s + (−0.955 + 0.294i)33-s + (0.988 + 0.149i)37-s + ⋯

Functional equation

Λ(s)=(392s/2ΓR(s+1)L(s)=((0.04270.999i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0427 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(392s/2ΓR(s+1)L(s)=((0.04270.999i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0427 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 392392    =    23722^{3} \cdot 7^{2}
Sign: 0.04270.999i-0.0427 - 0.999i
Analytic conductor: 42.126242.1262
Root analytic conductor: 42.126242.1262
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ392(285,)\chi_{392} (285, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 392, (1: ), 0.04270.999i)(1,\ 392,\ (1:\ ),\ -0.0427 - 0.999i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.68034275330.7100621476i0.6803427533 - 0.7100621476i
L(12)L(\frac12) \approx 0.68034275330.7100621476i0.6803427533 - 0.7100621476i
L(1)L(1) \approx 1.032718105+0.07274365166i1.032718105 + 0.07274365166i
L(1)L(1) \approx 1.032718105+0.07274365166i1.032718105 + 0.07274365166i

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
good3 1+(0.955+0.294i)T 1 + (0.955 + 0.294i)T
5 1+(0.733+0.680i)T 1 + (-0.733 + 0.680i)T
11 1+(0.826+0.563i)T 1 + (-0.826 + 0.563i)T
13 1+(0.9000.433i)T 1 + (-0.900 - 0.433i)T
17 1+(0.3650.930i)T 1 + (-0.365 - 0.930i)T
19 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
23 1+(0.3650.930i)T 1 + (0.365 - 0.930i)T
29 1+(0.623+0.781i)T 1 + (-0.623 + 0.781i)T
31 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
37 1+(0.988+0.149i)T 1 + (0.988 + 0.149i)T
41 1+(0.222+0.974i)T 1 + (0.222 + 0.974i)T
43 1+(0.2220.974i)T 1 + (0.222 - 0.974i)T
47 1+(0.07470.997i)T 1 + (-0.0747 - 0.997i)T
53 1+(0.9880.149i)T 1 + (0.988 - 0.149i)T
59 1+(0.7330.680i)T 1 + (-0.733 - 0.680i)T
61 1+(0.9880.149i)T 1 + (-0.988 - 0.149i)T
67 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
71 1+(0.623+0.781i)T 1 + (0.623 + 0.781i)T
73 1+(0.0747+0.997i)T 1 + (-0.0747 + 0.997i)T
79 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
83 1+(0.900+0.433i)T 1 + (-0.900 + 0.433i)T
89 1+(0.8260.563i)T 1 + (-0.826 - 0.563i)T
97 1T 1 - 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

−24.32569823829457606669070214255, −23.890294364217125712400107063208, −22.94852095625988070346539524684, −21.40449217972606616826388985054, −21.075196952865455299948155853918, −19.87274427596621910052396979072, −19.37417103956034881019573753925, −18.66424865356228416610773772457, −17.401236094232942525037645702500, −16.41798582445063492773953376982, −15.47596387217556112720457630143, −14.807740739952319075412642334805, −13.702624952224089191945761488125, −12.85107897866236963981929950114, −12.181633068402145709998433483921, −10.964108534260790000444265173919, −9.7568616391741779502001063654, −8.80690690652337987220427053568, −7.99803231294197944412544023971, −7.36298549520942760628970661329, −5.93799405308837629906364904198, −4.57888460502474408550866419964, −3.71407262955854739303469639222, −2.53197098480124441459900202335, −1.298583149487686324574490623322, 0.23470224434600038651792229127, 2.40576387064497229843928732189, 2.8665908927099713649924344920, 4.23318507138911775711811437962, 5.01092283770255809302903820637, 6.85799027297320002989798592919, 7.49831870292937420569849755207, 8.370744811124707058502618431331, 9.50659677976628763963701476157, 10.37276193718787323559756744202, 11.24592593901866821785960374003, 12.51977255790164820493661064315, 13.343556417018077204386998640638, 14.485167960505510344301308692678, 15.15160273099884249872663883000, 15.68487173351106757779538433587, 16.844269469076334241897774542835, 18.237455082151163479241012752769, 18.730057552480973434291833174888, 19.94949168627893550722555915113, 20.18120806189071127716387678678, 21.39309837176308404186420947904, 22.25533892978246645813890123136, 23.05417396729178609325645054616, 24.11696398059342280596788507131

Graph of the ZZ-function along the critical line