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Properties

Label 1-279-279.148-r1-0-0
Degree $1$
Conductor $279$
Sign $-0.733 - 0.679i$
Analytic cond. $29.9827$
Root an. cond. $29.9827$
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  + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.5 + 0.866i)5-s + (−0.978 + 0.207i)7-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)10-s + (−0.309 + 0.951i)11-s + (−0.913 + 0.406i)13-s + (0.309 + 0.951i)14-s + (0.913 − 0.406i)16-s + (0.978 + 0.207i)17-s + (−0.104 − 0.994i)19-s + (0.309 − 0.951i)20-s + (0.978 + 0.207i)22-s + (−0.669 + 0.743i)23-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.5 + 0.866i)5-s + (−0.978 + 0.207i)7-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)10-s + (−0.309 + 0.951i)11-s + (−0.913 + 0.406i)13-s + (0.309 + 0.951i)14-s + (0.913 − 0.406i)16-s + (0.978 + 0.207i)17-s + (−0.104 − 0.994i)19-s + (0.309 − 0.951i)20-s + (0.978 + 0.207i)22-s + (−0.669 + 0.743i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(279\)    =    \(3^{2} \cdot 31\)
Sign: $-0.733 - 0.679i$
Analytic conductor: \(29.9827\)
Root analytic conductor: \(29.9827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{279} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 279,\ (1:\ ),\ -0.733 - 0.679i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1238517648 - 0.3158646372i\)
\(L(\frac12)\) \(\approx\) \(0.1238517648 - 0.3158646372i\)
\(L(1)\) \(\approx\) \(0.6027870930 - 0.1387214577i\)
\(L(1)\) \(\approx\) \(0.6027870930 - 0.1387214577i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.104 - 0.994i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.978 + 0.207i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (-0.913 + 0.406i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.669 + 0.743i)T \)
29 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (-0.913 - 0.406i)T \)
47 \( 1 + (0.913 - 0.406i)T \)
53 \( 1 + (-0.669 + 0.743i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (0.978 - 0.207i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (0.669 + 0.743i)T \)
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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

−25.58090368751604986198624732345, −24.830239132843575833657878277, −24.05057367002108023377050029556, −23.1977045327034837841180471538, −22.4684729804077096645996648823, −21.351667365231448598653812427339, −20.112128220631833245409465110268, −19.18999218048462487100419543976, −18.499392950818972539704298031560, −17.00038034031847102809930133688, −16.54623018600401347516584258091, −15.85179736706124851638402283532, −14.79130207474597388318861165311, −13.69691606488017564670116835069, −12.82368401961616817730304552932, −12.003307559580939625643732329779, −10.21890234812705980204449022738, −9.52559890078035866861147068672, −8.24999614740648094595307605468, −7.73293223707285381188141096162, −6.33498981649459563723145666483, −5.47615367299821606513724571550, −4.32549837409261102113020282348, −3.205888833482612575578929956340, −0.84216801316580572949722507184, 0.14971681032378394077208737653, 2.103649462535478496982761890946, 3.044802699618420649496369678750, 4.020752049162199793406274263349, 5.316874036319793249411499054610, 6.88117740362796073323730619490, 7.75276374851360483712056006058, 9.25278450736375506972687428676, 9.97202753789119193125352343314, 10.824331533024489920897717679434, 12.04873053165583576359273785593, 12.518048010210673458992030034380, 13.7710100982062211056916526605, 14.74297695836563433002843184329, 15.70958334731658964612538332314, 17.01281500243096153108127825449, 18.02468001166285097623612488375, 18.87810215354450979457118816089, 19.58010354984089814796437402516, 20.244736603561704837325648880588, 21.735972600305666538746952742875, 22.061485317664013049107994030804, 23.14114804446847851421499976787, 23.66756409018310951074699264018, 25.498432085985231697231095024629

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