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Properties

Label 1-1368-1368.43-r1-0-0
Degree $1$
Conductor $1368$
Sign $-0.977 - 0.211i$
Analytic cond. $147.012$
Root an. cond. $147.012$
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  + (0.939 + 0.342i)5-s − 7-s + (−0.5 + 0.866i)11-s + (−0.173 + 0.984i)13-s + (0.766 − 0.642i)17-s + (−0.766 − 0.642i)23-s + (0.766 + 0.642i)25-s + (−0.173 + 0.984i)29-s + (0.5 + 0.866i)31-s + (−0.939 − 0.342i)35-s − 37-s + (0.766 − 0.642i)41-s + (0.766 − 0.642i)43-s + (−0.173 + 0.984i)47-s + 49-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)5-s − 7-s + (−0.5 + 0.866i)11-s + (−0.173 + 0.984i)13-s + (0.766 − 0.642i)17-s + (−0.766 − 0.642i)23-s + (0.766 + 0.642i)25-s + (−0.173 + 0.984i)29-s + (0.5 + 0.866i)31-s + (−0.939 − 0.342i)35-s − 37-s + (0.766 − 0.642i)41-s + (0.766 − 0.642i)43-s + (−0.173 + 0.984i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.977 - 0.211i$
Analytic conductor: \(147.012\)
Root analytic conductor: \(147.012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1368,\ (1:\ ),\ -0.977 - 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06297558906 + 0.5876762126i\)
\(L(\frac12)\) \(\approx\) \(-0.06297558906 + 0.5876762126i\)
\(L(1)\) \(\approx\) \(0.9353481559 + 0.2209501559i\)
\(L(1)\) \(\approx\) \(0.9353481559 + 0.2209501559i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.173 + 0.984i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + (-0.173 + 0.984i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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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

−20.35062352144482400151377831564, −19.3823005005942584093827232055, −18.89091587957079932482978012811, −17.86118884736005846126175260213, −17.2791217777877181394655503280, −16.4328767117006178649566631985, −15.86622566475022014464533955005, −14.943741189591413396791055914963, −13.96060392692021852928509915864, −13.24898767402131810866190894739, −12.828372845237081361885193063607, −11.900707445409042213952646517551, −10.772257459955693082189579748134, −9.95882047792148190115239288963, −9.62788156616307883383195074429, −8.444483800939332713287605541933, −7.836831781563713009591721854, −6.59585350649273470610132074349, −5.74319674178324567212084085876, −5.49040198081946519212629557721, −4.03634303999365924340829374956, −3.11680510172789823921893626019, −2.334673398265503007843255256247, −1.06640531686350529658822222238, −0.11576265809651295077722039762, 1.375576424292756434404376778544, 2.37781396819999145672942001672, 3.067812020884793784551474977326, 4.240408575126839998382555369974, 5.22616505618121858977795657952, 6.053727459623149342525150016270, 6.89716317476968134364772621726, 7.41814629447990385493125361594, 8.85949520710324653926356945060, 9.4578993961452487125202010466, 10.186481699972995141618046477618, 10.72639439480327283144522959034, 12.16715984778779977306294701729, 12.46970616722793338848221561166, 13.56234903881842188676053777119, 14.10088909798972061259087664353, 14.8433732324275236964501550916, 15.98284050565181415174475459423, 16.38449013910467726480471976898, 17.38736910468167249659390408840, 18.03021865637008931483810095101, 18.82325352407206619314418523329, 19.39343767444397769445975448497, 20.529832510159693533964122246438, 20.95786778685296663434602671421

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