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Properties

Label 1-504-504.419-r1-0-0
Degree $1$
Conductor $504$
Sign $-0.342 - 0.939i$
Analytic cond. $54.1623$
Root an. cond. $54.1623$
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  + (0.5 + 0.866i)5-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + 17-s − 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s − 37-s + (−0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s + (0.5 − 0.866i)47-s + 53-s + 55-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + 17-s − 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s − 37-s + (−0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s + (0.5 − 0.866i)47-s + 53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(54.1623\)
Root analytic conductor: \(54.1623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 504,\ (1:\ ),\ -0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6386211622 - 0.9120455399i\)
\(L(\frac12)\) \(\approx\) \(0.6386211622 - 0.9120455399i\)
\(L(1)\) \(\approx\) \(1.006770536 - 0.06859741530i\)
\(L(1)\) \(\approx\) \(1.006770536 - 0.06859741530i\)

Euler product

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

−23.72179474914528300280245485325, −22.95438050063064486869695680654, −21.752752728427394228000706445270, −21.257830490255203018650482631006, −20.31370813795409932964128276464, −19.55848275256119330591908635073, −18.648741817288400618453996060042, −17.42790192983301355050248448848, −17.03538918499010177512717101564, −16.14751676578813150190986446446, −15.04299150560947575341266767792, −14.19942945365547281318950973950, −13.321715647007429506673480797248, −12.29389637663505171704749916932, −11.843776435305668669896429019604, −10.36288969069448125003567829222, −9.576751238621541867900385250932, −8.86611549675032213123237719452, −7.73637947198136227661285860084, −6.71558182184660267692385496745, −5.64162370059132072395613069892, −4.70172524856304859009518842494, −3.80243504566543033946593927459, −2.15469481476684266466525440037, −1.38120887374165865805071836171, 0.26791472652876571823968400940, 1.815245979168723931804242583345, 2.95026739249836834419047174923, 3.804108788832768362265173302130, 5.32962647154689778865484847527, 6.09978745990720329080968270868, 7.04363538364198062009467740598, 8.07204647829293422188946328909, 9.09857377102081000277242404374, 10.24035998061110860366993520407, 10.71596733048843865441489078123, 11.84751291127803005996617201494, 12.806753424394334562234078017004, 13.8186990004284658947953844396, 14.58827712381673442836551501531, 15.20843338084404098343759249419, 16.55344399961001450978560410287, 17.128156845936632697743203516754, 18.21432302395548080562472297487, 18.83045323843258640168963924050, 19.6820906654427937071390832646, 20.73061398527923965836302229760, 21.64447851187583896404730552787, 22.262310515316638171471277063009, 23.0054215589493257066008886530

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