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Properties

Label 1-1480-1480.1419-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.978 - 0.205i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.978 - 0.205i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (1419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.978 - 0.205i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01413533966 - 0.1363893519i\)
\(L(\frac12)\) \(\approx\) \(0.01413533966 - 0.1363893519i\)
\(L(1)\) \(\approx\) \(0.6511298190 + 0.01770305048i\)
\(L(1)\) \(\approx\) \(0.6511298190 + 0.01770305048i\)

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

−21.55754950772083414056949018960, −20.08816563538560170435994682104, −19.39327679576268867783953973632, −18.63816901306789994406057675217, −18.02322584259099385389335567200, −17.290540544729909885822858137406, −16.619229152908274225017557167173, −15.807341506220385572880923778938, −15.07338534451446596785554949634, −14.2367140662310001090142146222, −13.17948754425312557030426620297, −12.55208022134497285788682276407, −11.826439510781358039149186933440, −11.25566513406418490076605526149, −10.33888491761860335729236583759, −9.59730919933766291131984085014, −8.35628021200937753392456804272, −7.97974382547926749143229438927, −6.66247080050495761421424585343, −6.04571435208424772367211711265, −5.44550072993214540462444034380, −4.51955847406330323609634437600, −3.42763608410950654444441954419, −2.2360236697842150796120062445, −1.391943968693907920100006943286, 0.06116688435498746323296181004, 1.3013737053473369708986311664, 2.39026017864687116135568441827, 3.8337509816789683705181837077, 4.59827834774779735930949390369, 4.918016264100677642507751835612, 6.40536279404903704162879323889, 6.844815202233352819604503562240, 7.544030642762439876287706019358, 8.91100737031096838266423993198, 9.65336624344276045158279967813, 10.391721193844327677381413816514, 11.053342335050269641816617522133, 11.9632216923498739998685228346, 12.38347887856496093503887632597, 13.58471193384518695661774586455, 14.17552050578562703199822934110, 15.18206553484824326771133681297, 15.87436764760076931176077540278, 16.69280506951284800676141122943, 17.34271450640264537674728462877, 17.7517246347444667309708299100, 18.67649484118705296768079802602, 19.82635005903844612001604343187, 20.17148874367082387955595876756

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