Properties

Label 1-1480-1480.579-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.877 + 0.479i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
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)3-s + (0.173 − 0.984i)7-s + (0.766 − 0.642i)9-s + (0.5 + 0.866i)11-s + (0.642 − 0.766i)13-s + (0.642 + 0.766i)17-s + (0.342 + 0.939i)19-s + (0.173 + 0.984i)21-s + (0.866 + 0.5i)23-s + (−0.5 + 0.866i)27-s + (−0.866 + 0.5i)29-s i·31-s + (−0.766 − 0.642i)33-s + (−0.342 + 0.939i)39-s + (−0.766 − 0.642i)41-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)3-s + (0.173 − 0.984i)7-s + (0.766 − 0.642i)9-s + (0.5 + 0.866i)11-s + (0.642 − 0.766i)13-s + (0.642 + 0.766i)17-s + (0.342 + 0.939i)19-s + (0.173 + 0.984i)21-s + (0.866 + 0.5i)23-s + (−0.5 + 0.866i)27-s + (−0.866 + 0.5i)29-s i·31-s + (−0.766 − 0.642i)33-s + (−0.342 + 0.939i)39-s + (−0.766 − 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.877 + 0.479i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.877 + 0.479i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.225785174 + 0.3129096455i\)
\(L(\frac12)\) \(\approx\) \(1.225785174 + 0.3129096455i\)
\(L(1)\) \(\approx\) \(0.9211200922 + 0.08253910182i\)
\(L(1)\) \(\approx\) \(0.9211200922 + 0.08253910182i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.642 - 0.766i)T \)
17 \( 1 + (0.642 + 0.766i)T \)
19 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (0.984 - 0.173i)T \)
61 \( 1 + (0.642 - 0.766i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + T \)
79 \( 1 + (0.984 + 0.173i)T \)
83 \( 1 + (-0.766 + 0.642i)T \)
89 \( 1 + (-0.984 + 0.173i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−20.88772215640200469108846330340, −19.6040160816275253935433249050, −18.819323131582977214020100857, −18.482090133555898584080276015074, −17.67569419817833997181694336353, −16.745530132102181040772678190571, −16.258457793344774755833924569210, −15.5007486118431802898363366004, −14.5317854546716131113618993976, −13.634820208648816243194908972680, −12.98015693498420281553467193489, −11.91012104612135054538757951429, −11.566704466745174775654878139853, −10.945660892237814871390994417033, −9.80496359372173931757418744408, −8.94176859451652140243319503107, −8.28206257801940062474397706476, −6.99472121423202928638349167863, −6.56930124193386340281523887447, −5.47630974542512038231971691490, −5.12711901551354338552171749220, −3.89633803921577077041623554880, −2.80524822466908775687891233730, −1.71763351593430013732681783640, −0.729176410811865143579576943899, 0.963929963962391626381474936704, 1.65491190530992325619798996082, 3.48389026094439467018993710912, 3.92542857086579040064062611331, 4.943341971299156526252755835925, 5.70953094050989096465430213595, 6.56141845282822842066760023830, 7.398728409399308988711355828172, 8.11644988535501321037914987648, 9.500334503384682560542638267526, 9.98375689369863489231354230839, 10.85223273515328094667616038893, 11.32213877721691656505508614049, 12.441913424695836395292342537017, 12.86601657392076189350194882897, 13.91205205486733695616598962287, 14.87444528383423783371675102974, 15.3886086873897667255014223021, 16.50707145687196647196119643525, 16.91496771268058271179813464635, 17.58057119843244532003460138002, 18.28548943628308477968016563996, 19.160602493359416544807530716357, 20.21863296269416427804160358905, 20.71258875365009781412719607139

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