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Properties

Label 1-1259-1259.1000-r1-0-0
Degree $1$
Conductor $1259$
Sign $0.799 + 0.600i$
Analytic cond. $135.298$
Root an. cond. $135.298$
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  + (0.405 + 0.914i)2-s + (−0.0174 + 0.999i)3-s + (−0.671 + 0.740i)4-s + (0.961 − 0.276i)5-s + (−0.921 + 0.389i)6-s + (0.472 + 0.881i)7-s + (−0.949 − 0.314i)8-s + (−0.999 − 0.0349i)9-s + (0.641 + 0.766i)10-s + (−0.598 + 0.801i)11-s + (−0.728 − 0.684i)12-s + (−0.418 − 0.908i)13-s + (−0.614 + 0.788i)14-s + (0.259 + 0.965i)15-s + (−0.0972 − 0.995i)16-s + (−0.902 − 0.430i)17-s + ⋯
L(s)  = 1  + (0.405 + 0.914i)2-s + (−0.0174 + 0.999i)3-s + (−0.671 + 0.740i)4-s + (0.961 − 0.276i)5-s + (−0.921 + 0.389i)6-s + (0.472 + 0.881i)7-s + (−0.949 − 0.314i)8-s + (−0.999 − 0.0349i)9-s + (0.641 + 0.766i)10-s + (−0.598 + 0.801i)11-s + (−0.728 − 0.684i)12-s + (−0.418 − 0.908i)13-s + (−0.614 + 0.788i)14-s + (0.259 + 0.965i)15-s + (−0.0972 − 0.995i)16-s + (−0.902 − 0.430i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $0.799 + 0.600i$
Analytic conductor: \(135.298\)
Root analytic conductor: \(135.298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1000, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (1:\ ),\ 0.799 + 0.600i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.483256385 + 0.4945005614i\)
\(L(\frac12)\) \(\approx\) \(1.483256385 + 0.4945005614i\)
\(L(1)\) \(\approx\) \(0.8670174799 + 0.8154241131i\)
\(L(1)\) \(\approx\) \(0.8670174799 + 0.8154241131i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (0.405 + 0.914i)T \)
3 \( 1 + (-0.0174 + 0.999i)T \)
5 \( 1 + (0.961 - 0.276i)T \)
7 \( 1 + (0.472 + 0.881i)T \)
11 \( 1 + (-0.598 + 0.801i)T \)
13 \( 1 + (-0.418 - 0.908i)T \)
17 \( 1 + (-0.902 - 0.430i)T \)
19 \( 1 + (-0.498 - 0.866i)T \)
23 \( 1 + (0.690 + 0.723i)T \)
29 \( 1 + (0.594 - 0.803i)T \)
31 \( 1 + (-0.532 - 0.846i)T \)
37 \( 1 + (-0.0274 - 0.999i)T \)
41 \( 1 + (0.967 - 0.251i)T \)
43 \( 1 + (-0.117 + 0.993i)T \)
47 \( 1 + (-0.732 + 0.681i)T \)
53 \( 1 + (0.718 - 0.695i)T \)
59 \( 1 + (-0.802 + 0.596i)T \)
61 \( 1 + (0.765 + 0.643i)T \)
67 \( 1 + (-0.377 - 0.926i)T \)
71 \( 1 + (0.990 - 0.139i)T \)
73 \( 1 + (-0.735 - 0.677i)T \)
79 \( 1 + (0.181 + 0.983i)T \)
83 \( 1 + (-0.602 - 0.798i)T \)
89 \( 1 + (0.459 - 0.888i)T \)
97 \( 1 + (-0.999 - 0.0149i)T \)
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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

−20.79057124885581431065524800746, −20.073018774757010410110455048478, −19.22418482153687816690089878093, −18.60674196849631047878077514498, −17.97537860373994151436715689202, −17.21615717015172008512862844241, −16.53492921804915936516713042023, −14.86705253432140207472746830878, −14.210857153791695900993312938444, −13.78259716294574236321073905131, −13.0542640385168818497368557380, −12.44606629256127697141595709275, −11.33269963259253000028131801465, −10.78143505037736396105388236162, −10.1471954838075705981849024628, −8.90494155214873469335222807682, −8.35154615345368445773250269681, −6.97745673015506897289179785536, −6.41321506899119234094175827525, −5.44295450106135337062941575815, −4.62333118414420863876061085011, −3.41841962314089852939177747010, −2.45777167626717507502131689261, −1.74829344350500667980197107354, −0.95937687792399412518271654387, 0.28041529411405397323395964754, 2.33064264522026723022612799782, 2.85026012166489222806471316121, 4.370367441615048201801386574205, 4.93518282362665865609323015478, 5.51473264121043435157153597601, 6.24693901888753635085472180462, 7.453455076308392279537444498395, 8.36785424127197462672182172137, 9.23052706185164561378156295081, 9.573765725915055984992955881879, 10.698726187557087257723130039115, 11.64103174657483140093007542448, 12.759706900835940239653496354591, 13.22921130610750619932722783748, 14.27089804220475480660215226367, 15.05137205446664850816804302095, 15.39740283030171246001087932552, 16.15916823986761073630215495624, 17.1796907721415933554703424170, 17.766312895612637318687563703014, 18.020936318605271428216902536357, 19.56074432859222408180731983505, 20.57981246102287588002336301670, 21.296251279084967315828668382195

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