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Properties

Label 1-1368-1368.299-r1-0-0
Degree $1$
Conductor $1368$
Sign $-0.546 - 0.837i$
Analytic cond. $147.012$
Root an. cond. $147.012$
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)5-s − 7-s + (0.5 − 0.866i)11-s + (0.173 − 0.984i)13-s + (−0.766 + 0.642i)17-s + (0.766 + 0.642i)23-s + (0.766 + 0.642i)25-s + (−0.173 + 0.984i)29-s + (−0.5 − 0.866i)31-s + (0.939 + 0.342i)35-s + 37-s + (0.766 − 0.642i)41-s + (0.766 − 0.642i)43-s + (0.173 − 0.984i)47-s + 49-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)5-s − 7-s + (0.5 − 0.866i)11-s + (0.173 − 0.984i)13-s + (−0.766 + 0.642i)17-s + (0.766 + 0.642i)23-s + (0.766 + 0.642i)25-s + (−0.173 + 0.984i)29-s + (−0.5 − 0.866i)31-s + (0.939 + 0.342i)35-s + 37-s + (0.766 − 0.642i)41-s + (0.766 − 0.642i)43-s + (0.173 − 0.984i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.546 - 0.837i$
Analytic conductor: \(147.012\)
Root analytic conductor: \(147.012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1368,\ (1:\ ),\ -0.546 - 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4731856533 - 0.8733377338i\)
\(L(\frac12)\) \(\approx\) \(0.4731856533 - 0.8733377338i\)
\(L(1)\) \(\approx\) \(0.7861667824 - 0.1651390423i\)
\(L(1)\) \(\approx\) \(0.7861667824 - 0.1651390423i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 - T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.173 + 0.984i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (-0.173 + 0.984i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (0.939 + 0.342i)T \)
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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

−20.77019296637017451231562390187, −20.039098668997990593221524958267, −19.37152591935795381120629220673, −18.86095926944738796734493327571, −17.99265328805626538625218639813, −17.064629304734602092225038680715, −16.11647063115879647086129262250, −15.83095200515433671951860692433, −14.777669983311035567010370677, −14.25284597200515002896842530012, −13.04835918343251201995620995075, −12.55153021886595109859020055926, −11.53746195100462880452584861095, −11.12152843938045748567717753992, −9.90944075405529159050896106150, −9.30631895788104511907345216993, −8.47354365309199498987537253282, −7.26991063139050135109778193147, −6.87631677924568511581664319152, −6.08918118859740926538208438076, −4.58317917215155546372944627314, −4.16633825622745071556171345085, −3.10288381448854276446635250201, −2.26821601958788036123892469384, −0.8338935074893167704659425484, 0.2850202606655165812672337569, 1.06004871331408725757521541707, 2.63353168520036439440695640222, 3.56310503467408389590735369471, 4.02890514418155879933798174775, 5.3380752286029971950662922234, 6.077555875949423686609361510832, 7.04199893520773531734298637708, 7.815787937951987515115350271749, 8.80422859405198059130696785797, 9.224623838898409004318718869651, 10.50646913898909833372202964489, 11.08661369827411729260351711860, 11.94539347535273303753733720956, 12.93357270080239321812775051243, 13.1432326047955985664377839999, 14.372214442493616960492176229607, 15.34678678158481445918978640898, 15.72464434570616432700710179954, 16.63478160050642167980126612181, 17.1358794647792817437889241483, 18.31720574548453355458233472134, 19.03422308224235258259991305996, 19.768382291306704262447785985813, 20.07198177633981190719216729207

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