Properties

Label 1-18e2-324.203-r0-0-0
Degree $1$
Conductor $324$
Sign $0.154 + 0.987i$
Analytic cond. $1.50464$
Root an. cond. $1.50464$
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  + (−0.893 − 0.448i)5-s + (0.286 + 0.957i)7-s + (−0.0581 + 0.998i)11-s + (0.597 − 0.802i)13-s + (−0.766 + 0.642i)17-s + (−0.766 − 0.642i)19-s + (−0.286 + 0.957i)23-s + (0.597 + 0.802i)25-s + (−0.396 + 0.918i)29-s + (0.686 + 0.727i)31-s + (0.173 − 0.984i)35-s + (0.173 + 0.984i)37-s + (0.993 + 0.116i)41-s + (0.835 + 0.549i)43-s + (−0.686 + 0.727i)47-s + ⋯
L(s)  = 1  + (−0.893 − 0.448i)5-s + (0.286 + 0.957i)7-s + (−0.0581 + 0.998i)11-s + (0.597 − 0.802i)13-s + (−0.766 + 0.642i)17-s + (−0.766 − 0.642i)19-s + (−0.286 + 0.957i)23-s + (0.597 + 0.802i)25-s + (−0.396 + 0.918i)29-s + (0.686 + 0.727i)31-s + (0.173 − 0.984i)35-s + (0.173 + 0.984i)37-s + (0.993 + 0.116i)41-s + (0.835 + 0.549i)43-s + (−0.686 + 0.727i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.154 + 0.987i$
Analytic conductor: \(1.50464\)
Root analytic conductor: \(1.50464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 324,\ (0:\ ),\ 0.154 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6735971542 + 0.5764368281i\)
\(L(\frac12)\) \(\approx\) \(0.6735971542 + 0.5764368281i\)
\(L(1)\) \(\approx\) \(0.8562784339 + 0.1886973804i\)
\(L(1)\) \(\approx\) \(0.8562784339 + 0.1886973804i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.893 - 0.448i)T \)
7 \( 1 + (0.286 + 0.957i)T \)
11 \( 1 + (-0.0581 + 0.998i)T \)
13 \( 1 + (0.597 - 0.802i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (-0.286 + 0.957i)T \)
29 \( 1 + (-0.396 + 0.918i)T \)
31 \( 1 + (0.686 + 0.727i)T \)
37 \( 1 + (0.173 + 0.984i)T \)
41 \( 1 + (0.993 + 0.116i)T \)
43 \( 1 + (0.835 + 0.549i)T \)
47 \( 1 + (-0.686 + 0.727i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.0581 - 0.998i)T \)
61 \( 1 + (0.973 + 0.230i)T \)
67 \( 1 + (-0.396 - 0.918i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (0.993 - 0.116i)T \)
83 \( 1 + (-0.993 + 0.116i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (0.893 - 0.448i)T \)
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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

−24.632698558512084622630618217580, −23.98033431204966962570364792969, −23.12412682295713024709166476188, −22.46854122497944774040741818987, −21.19276357302633411081257758178, −20.49360267635395488079979124885, −19.37289150081835274438831503358, −18.83066262794534335587583877828, −17.7629668421826772259461780840, −16.53347327914949989832765834329, −16.06046578206727844831522656259, −14.80461508483767371547668186750, −14.014015892866702347211431805957, −13.14601204371670830029255055633, −11.72902046967513149951801996468, −11.11899344949655991301236764580, −10.311472660066913107301860767230, −8.82300168723696760711120222697, −7.98392511549441237363839748490, −6.987592997643964262749497556704, −6.05891067525666631659025733708, −4.32646026588140028085979812769, −3.85499467034631699955999315752, −2.38099657277511636757657823316, −0.58915508885459151578089568832, 1.52717370352074120275446704189, 2.90324083262417361210491913239, 4.234330413397978604383664988200, 5.10494702811364156164091508452, 6.3253923659869021954110979860, 7.60935108448510611163817876439, 8.4611115323490070157852119073, 9.26061933608837287513273616326, 10.67147807231372562498426216042, 11.54365765573276446403954664991, 12.52746875104095137756269990807, 13.10052733016518400684883323195, 14.723346718891715695861318052485, 15.40874475914044449739096873082, 15.95176086310843026578274182073, 17.40394386980128347595565883799, 17.99566606418955230863980441430, 19.18667569191123429892625174812, 19.905822128848359321338721507245, 20.77272184189584414163088185815, 21.75157098358793341599277384246, 22.71565536071162298996140875668, 23.57423564244883689503898799425, 24.33564366851502126733146444684, 25.31332406506014229498948893445

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