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Properties

Label 1-3381-3381.470-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.512 - 0.858i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (0.862 − 0.505i)2-s + (0.488 − 0.872i)4-s + (0.917 + 0.396i)5-s + (−0.0203 − 0.999i)8-s + (0.992 − 0.122i)10-s + (0.685 − 0.728i)11-s + (0.947 + 0.320i)13-s + (−0.523 − 0.852i)16-s + (0.488 + 0.872i)17-s + (0.959 + 0.281i)19-s + (0.794 − 0.607i)20-s + (0.222 − 0.974i)22-s + (0.685 + 0.728i)25-s + (0.979 − 0.202i)26-s + (−0.488 − 0.872i)29-s + ⋯
L(s)  = 1  + (0.862 − 0.505i)2-s + (0.488 − 0.872i)4-s + (0.917 + 0.396i)5-s + (−0.0203 − 0.999i)8-s + (0.992 − 0.122i)10-s + (0.685 − 0.728i)11-s + (0.947 + 0.320i)13-s + (−0.523 − 0.852i)16-s + (0.488 + 0.872i)17-s + (0.959 + 0.281i)19-s + (0.794 − 0.607i)20-s + (0.222 − 0.974i)22-s + (0.685 + 0.728i)25-s + (0.979 − 0.202i)26-s + (−0.488 − 0.872i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.512 - 0.858i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.512 - 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.722396442 - 2.113749972i\)
\(L(\frac12)\) \(\approx\) \(3.722396442 - 2.113749972i\)
\(L(1)\) \(\approx\) \(2.151008613 - 0.7617300781i\)
\(L(1)\) \(\approx\) \(2.151008613 - 0.7617300781i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.862 - 0.505i)T \)
5 \( 1 + (0.917 + 0.396i)T \)
11 \( 1 + (0.685 - 0.728i)T \)
13 \( 1 + (0.947 + 0.320i)T \)
17 \( 1 + (0.488 + 0.872i)T \)
19 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 + (-0.488 - 0.872i)T \)
31 \( 1 + (0.415 - 0.909i)T \)
37 \( 1 + (-0.882 - 0.470i)T \)
41 \( 1 + (-0.917 - 0.396i)T \)
43 \( 1 + (-0.0203 + 0.999i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (-0.714 + 0.699i)T \)
59 \( 1 + (0.992 - 0.122i)T \)
61 \( 1 + (0.979 + 0.202i)T \)
67 \( 1 + (0.142 - 0.989i)T \)
71 \( 1 + (0.933 + 0.359i)T \)
73 \( 1 + (-0.818 + 0.574i)T \)
79 \( 1 + (-0.841 + 0.540i)T \)
83 \( 1 + (0.182 + 0.983i)T \)
89 \( 1 + (0.970 + 0.242i)T \)
97 \( 1 + (0.654 + 0.755i)T \)
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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

−18.80366212542333515707952066404, −17.843968823009851019141445295, −17.61128603674546041224773307644, −16.719133737973330792850780923562, −16.09320463510805681299621464491, −15.57282424307918064351658713689, −14.49327438541780153894877928839, −14.18256876940234149046462932631, −13.38694507846705535534875162247, −12.91012574976777302520323757730, −12.03975500950019564693710630483, −11.56213613454955435280148376892, −10.51267497178531503439628316196, −9.72925525351454227316513820293, −8.92561258018913780528055158737, −8.31111761735810140959528542244, −7.20657455071666323452500537215, −6.758307298031543961393933823337, −5.91200252056169323863349206443, −5.15644274114697851192058959555, −4.7664829456045787974637172979, −3.57785393274786627855216344187, −3.04082988587018754208136465176, −1.908819521056623485562305710444, −1.19989675538582452889129148305, 1.04447130587925073574529907582, 1.66567482899011837555127545790, 2.52714200621944569484841270105, 3.57293874571877422125074912381, 3.80468792891849697762603235648, 5.08689036022652359337529194162, 5.80456859916176807184400232221, 6.24302069125702855198799069182, 6.92999729388051725598966231930, 8.06358895475105779260990952273, 9.02460889754858211540773671558, 9.77621023303909508324890363376, 10.31592033791633397508342416444, 11.2353362337866034089495694857, 11.56118798502486196978802776302, 12.53115870313384628719710367269, 13.33075739837822086083754125029, 13.781881165825776988619235832927, 14.34120512883876539435361168979, 15.005908637715125343808152306, 15.836691337694300798137329185836, 16.59937708533163155914822785378, 17.27382954279723379880590256049, 18.231596003953995432737382624064, 18.88206480456318591267498805048

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