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Properties

Label 1-1380-1380.143-r0-0-0
Degree $1$
Conductor $1380$
Sign $0.778 - 0.627i$
Analytic cond. $6.40869$
Root an. cond. $6.40869$
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  + (0.755 + 0.654i)7-s + (−0.841 + 0.540i)11-s + (0.755 − 0.654i)13-s + (−0.909 − 0.415i)17-s + (−0.415 − 0.909i)19-s + (0.415 − 0.909i)29-s + (0.142 − 0.989i)31-s + (−0.281 − 0.959i)37-s + (0.959 + 0.281i)41-s + (0.989 − 0.142i)43-s − i·47-s + (0.142 + 0.989i)49-s + (−0.755 − 0.654i)53-s + (0.654 + 0.755i)59-s + (0.142 − 0.989i)61-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)7-s + (−0.841 + 0.540i)11-s + (0.755 − 0.654i)13-s + (−0.909 − 0.415i)17-s + (−0.415 − 0.909i)19-s + (0.415 − 0.909i)29-s + (0.142 − 0.989i)31-s + (−0.281 − 0.959i)37-s + (0.959 + 0.281i)41-s + (0.989 − 0.142i)43-s − i·47-s + (0.142 + 0.989i)49-s + (−0.755 − 0.654i)53-s + (0.654 + 0.755i)59-s + (0.142 − 0.989i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.778 - 0.627i$
Analytic conductor: \(6.40869\)
Root analytic conductor: \(6.40869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (0:\ ),\ 0.778 - 0.627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.407946496 - 0.4972194515i\)
\(L(\frac12)\) \(\approx\) \(1.407946496 - 0.4972194515i\)
\(L(1)\) \(\approx\) \(1.101976003 - 0.07215378191i\)
\(L(1)\) \(\approx\) \(1.101976003 - 0.07215378191i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (0.755 + 0.654i)T \)
11 \( 1 + (-0.841 + 0.540i)T \)
13 \( 1 + (0.755 - 0.654i)T \)
17 \( 1 + (-0.909 - 0.415i)T \)
19 \( 1 + (-0.415 - 0.909i)T \)
29 \( 1 + (0.415 - 0.909i)T \)
31 \( 1 + (0.142 - 0.989i)T \)
37 \( 1 + (-0.281 - 0.959i)T \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (0.989 - 0.142i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.755 - 0.654i)T \)
59 \( 1 + (0.654 + 0.755i)T \)
61 \( 1 + (0.142 - 0.989i)T \)
67 \( 1 + (0.540 - 0.841i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (-0.909 + 0.415i)T \)
79 \( 1 + (0.654 + 0.755i)T \)
83 \( 1 + (0.281 + 0.959i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (0.281 - 0.959i)T \)
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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

−20.95657670515739602450565010283, −20.31054391167348393986344111157, −19.33511308338513871076734795768, −18.66325183934751756787097417119, −17.83247954553913417648762029206, −17.25453502147546874532108985063, −16.213261064478178068690028490426, −15.82520245413777883593290117983, −14.64327854188273492244531286193, −14.079854377662330939067808316008, −13.338598082102791383967335348103, −12.55352742936417333480054104911, −11.51476514190117757876100483717, −10.756030624816302200188953124324, −10.39860546424143940029239006035, −9.04439143062792352053318465269, −8.39358721351589505828200494966, −7.675179113691564635756200733090, −6.68276039510514300966677529813, −5.89500659509468023056861392140, −4.833222871929851505248030734078, −4.13583237829048586857821758304, −3.16735023532942920358797821716, −1.97484835846033484839350398045, −1.09715068989443439093262369914, 0.644449725619889904203589971278, 2.1622260535066902014242073803, 2.572072718397262361541196954294, 3.97647391207482042921844494249, 4.83145643342760832261381295880, 5.56103627706996648789701597520, 6.46892421650578017555031320978, 7.52768282262506768574630060098, 8.223351462952074927019251782099, 8.97389902542991085152910709820, 9.85319554625272202635235833903, 10.95681921176088570939459541999, 11.26567849867978006259614237134, 12.38745736842686367378677304123, 13.07787848798490864272972634812, 13.79502228053316766146767332077, 14.82115478373655088069512104580, 15.54997859300450440512264229779, 15.84280259605154965366598928593, 17.215339823702243179068858444510, 17.84378389169082208866484795803, 18.27974727592112960106856864597, 19.21286070153985192959857319516, 20.097915732233861801082816945317, 20.866137654963675803542532139985

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