Properties

Label 1-18e2-324.23-r0-0-0
Degree $1$
Conductor $324$
Sign $-0.483 - 0.875i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.396 − 0.918i)5-s + (0.0581 − 0.998i)7-s + (0.597 + 0.802i)11-s + (−0.686 − 0.727i)13-s + (−0.173 − 0.984i)17-s + (−0.173 + 0.984i)19-s + (−0.0581 − 0.998i)23-s + (−0.686 + 0.727i)25-s + (−0.973 − 0.230i)29-s + (−0.893 − 0.448i)31-s + (−0.939 + 0.342i)35-s + (−0.939 − 0.342i)37-s + (0.286 − 0.957i)41-s + (0.993 − 0.116i)43-s + (0.893 − 0.448i)47-s + ⋯
L(s)  = 1  + (−0.396 − 0.918i)5-s + (0.0581 − 0.998i)7-s + (0.597 + 0.802i)11-s + (−0.686 − 0.727i)13-s + (−0.173 − 0.984i)17-s + (−0.173 + 0.984i)19-s + (−0.0581 − 0.998i)23-s + (−0.686 + 0.727i)25-s + (−0.973 − 0.230i)29-s + (−0.893 − 0.448i)31-s + (−0.939 + 0.342i)35-s + (−0.939 − 0.342i)37-s + (0.286 − 0.957i)41-s + (0.993 − 0.116i)43-s + (0.893 − 0.448i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.483 - 0.875i$
Analytic conductor: \(1.50464\)
Root analytic conductor: \(1.50464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 324,\ (0:\ ),\ -0.483 - 0.875i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4597063523 - 0.7786989588i\)
\(L(\frac12)\) \(\approx\) \(0.4597063523 - 0.7786989588i\)
\(L(1)\) \(\approx\) \(0.8298791771 - 0.3530774015i\)
\(L(1)\) \(\approx\) \(0.8298791771 - 0.3530774015i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−25.56895902187653698150095833642, −24.277877916804450382901559029094, −23.852417878453775478548408151456, −22.45682909719133330095830380951, −21.898215679954598852070564654, −21.302677255855910122651515381018, −19.54174600833493690224186688390, −19.33541255500403301333001449587, −18.31567775128575383177550565008, −17.39503012645718813975179475841, −16.28664842977840988763489101692, −15.23200851131682335054587750065, −14.687435941094119266911254982468, −13.663180764888735738702586586105, −12.39107248775734055095127870003, −11.498483732700983213395848602006, −10.85473996874140880470325076761, −9.45512271870117068336665694124, −8.6759431532294216868384968758, −7.4358448640827970290631618438, −6.48372887844765159133845363368, −5.5186491975167916025072758198, −4.070599866464085184176609556748, −3.017963625707800281601884587819, −1.88214327813301563429979874282, 0.56596231827964708703131926401, 2.00057550654028831104890940366, 3.73966588617695457381455668097, 4.50949504813705237157080794672, 5.56577412129909553607650300075, 7.1315544154734861825849887085, 7.71980362224953410377977196406, 8.978685116037461448825017046729, 9.885032764759965861211985150098, 10.91093117407969452761861996997, 12.16764427864979107355971822908, 12.697330329308836165673719923844, 13.87184475135390053323472895577, 14.7788342138281743216780369902, 15.85455184648725284899251829310, 16.85574088897020400699286749744, 17.29079781378686697151027371463, 18.5478180300968675258383090176, 19.74858770719615785568321731575, 20.35043724129739312405588236697, 20.85810451154754033165805967207, 22.513633838623398259556551699794, 22.8669313305641597357359258474, 24.0505664615718976480740990057, 24.6830041506586782181390263470

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