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Properties

Label 1-12e3-1728.115-r1-0-0
Degree $1$
Conductor $1728$
Sign $-0.0617 + 0.998i$
Analytic cond. $185.699$
Root an. cond. $185.699$
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  + (−0.737 + 0.675i)5-s + (0.819 + 0.573i)7-s + (−0.0436 − 0.999i)11-s + (0.461 + 0.887i)13-s + (−0.866 − 0.5i)17-s + (−0.130 − 0.991i)19-s + (−0.819 + 0.573i)23-s + (0.0871 − 0.996i)25-s + (0.300 + 0.953i)29-s + (0.173 − 0.984i)31-s + (−0.991 + 0.130i)35-s + (0.130 − 0.991i)37-s + (0.0871 + 0.996i)41-s + (0.0436 + 0.999i)43-s + (0.984 − 0.173i)47-s + ⋯
L(s)  = 1  + (−0.737 + 0.675i)5-s + (0.819 + 0.573i)7-s + (−0.0436 − 0.999i)11-s + (0.461 + 0.887i)13-s + (−0.866 − 0.5i)17-s + (−0.130 − 0.991i)19-s + (−0.819 + 0.573i)23-s + (0.0871 − 0.996i)25-s + (0.300 + 0.953i)29-s + (0.173 − 0.984i)31-s + (−0.991 + 0.130i)35-s + (0.130 − 0.991i)37-s + (0.0871 + 0.996i)41-s + (0.0436 + 0.999i)43-s + (0.984 − 0.173i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0617 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0617 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.0617 + 0.998i$
Analytic conductor: \(185.699\)
Root analytic conductor: \(185.699\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1728,\ (1:\ ),\ -0.0617 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.063835474 + 1.131714676i\)
\(L(\frac12)\) \(\approx\) \(1.063835474 + 1.131714676i\)
\(L(1)\) \(\approx\) \(0.9660064026 + 0.1905907638i\)
\(L(1)\) \(\approx\) \(0.9660064026 + 0.1905907638i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.737 + 0.675i)T \)
7 \( 1 + (0.819 + 0.573i)T \)
11 \( 1 + (-0.0436 - 0.999i)T \)
13 \( 1 + (0.461 + 0.887i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.130 - 0.991i)T \)
23 \( 1 + (-0.819 + 0.573i)T \)
29 \( 1 + (0.300 + 0.953i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (0.130 - 0.991i)T \)
41 \( 1 + (0.0871 + 0.996i)T \)
43 \( 1 + (0.0436 + 0.999i)T \)
47 \( 1 + (0.984 - 0.173i)T \)
53 \( 1 + (-0.382 - 0.923i)T \)
59 \( 1 + (0.737 - 0.675i)T \)
61 \( 1 + (0.976 + 0.216i)T \)
67 \( 1 + (-0.887 + 0.461i)T \)
71 \( 1 + (0.258 + 0.965i)T \)
73 \( 1 + (-0.258 + 0.965i)T \)
79 \( 1 + (0.642 + 0.766i)T \)
83 \( 1 + (0.953 - 0.300i)T \)
89 \( 1 + (-0.965 - 0.258i)T \)
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

−20.13355125283366699215062623398, −19.32168546477522977389421389497, −18.35427345208467638262244482566, −17.56657954925192220704988152317, −17.11948429824873796504337163764, −16.148350088158151639534236429774, −15.41858460968675456279333065687, −14.898782836240376885338742499043, −13.90146209851233286398504338618, −13.170146016013617727719579113933, −12.227729717375206360847993046537, −11.91508644156123591804025623214, −10.60751974953018249638832866301, −10.42283369408900579262147634160, −9.129162842843082449842123818744, −8.23671734536613212304450923650, −7.89356649839993347345530153543, −6.98383996517634167399539599323, −5.92213724297691654439466835478, −4.92569610493370301615898690518, −4.27431214826522912684618306293, −3.65943105880360415235453062842, −2.22773620702989640298265906663, −1.351729564403965424706536094044, −0.36610271867391531852848124193, 0.80281471328786565662517515592, 2.096476186137777146316363607307, 2.8403795931968896655233868102, 3.89433122551813185686882356078, 4.58359339333369667817556832377, 5.63000535490340002479030934847, 6.49556548537482412348781758116, 7.2412077108542273947194378833, 8.19615985608302818040495595193, 8.72177655757517429227895811080, 9.590788068215827724668298508279, 10.85507860520239257064944242550, 11.4239369746310684702682532182, 11.58618597861513090292205368231, 12.83331591836159994121918892789, 13.775480294787996236101921393509, 14.34411076586063905542735938431, 15.11735221811343188353691649782, 15.90804037731764642893782244351, 16.313701589641151880198713859169, 17.61642952131055320620582163342, 18.11134679976466100536786360548, 18.828681898443497549191054546906, 19.45501617315395753025583983156, 20.20465031370306799373251985311

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