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Properties

Label 1-392-392.381-r1-0-0
Degree $1$
Conductor $392$
Sign $-0.0427 + 0.999i$
Analytic cond. $42.1262$
Root an. cond. $42.1262$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(1\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.0427 + 0.999i$
Analytic conductor: \(42.1262\)
Root analytic conductor: \(42.1262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 392,\ (1:\ ),\ -0.0427 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6803427533 + 0.7100621476i\)
\(L(\frac12)\) \(\approx\) \(0.6803427533 + 0.7100621476i\)
\(L(1)\) \(\approx\) \(1.032718105 - 0.07274365166i\)
\(L(1)\) \(\approx\) \(1.032718105 - 0.07274365166i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

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

Graph of the $Z$-function along the critical line