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Properties

Label 18-3343e9-3343.3342-c0e9-0-0
Degree $18$
Conductor $5.215\times 10^{31}$
Sign $1$
Analytic cond. $100.147$
Root an. cond. $1.29165$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 9·9-s − 11-s − 9·18-s − 19-s + 22-s + 9·25-s − 31-s + 38-s − 43-s + 9·49-s − 9·50-s − 53-s − 59-s − 61-s + 62-s + 45·81-s + 86-s − 89-s − 9·98-s − 9·99-s − 103-s + 106-s − 107-s − 109-s + 118-s + 122-s + ⋯
L(s)  = 1  − 2-s + 9·9-s − 11-s − 9·18-s − 19-s + 22-s + 9·25-s − 31-s + 38-s − 43-s + 9·49-s − 9·50-s − 53-s − 59-s − 61-s + 62-s + 45·81-s + 86-s − 89-s − 9·98-s − 9·99-s − 103-s + 106-s − 107-s − 109-s + 118-s + 122-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3343^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3343^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(3343^{9}\)
Sign: $1$
Analytic conductor: \(100.147\)
Root analytic conductor: \(1.29165\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3343} (3342, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((18,\ 3343^{9} ,\ ( \ : [0]^{9} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(7.270154845\)
\(L(\frac12)\) \(\approx\) \(7.270154845\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3343 \( 1+O(T) \)
good2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
3 \( ( 1 - T )^{9}( 1 + T )^{9} \)
5 \( ( 1 - T )^{9}( 1 + T )^{9} \)
7 \( ( 1 - T )^{9}( 1 + T )^{9} \)
11 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
13 \( ( 1 - T )^{9}( 1 + T )^{9} \)
17 \( ( 1 - T )^{9}( 1 + T )^{9} \)
19 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
23 \( ( 1 - T )^{9}( 1 + T )^{9} \)
29 \( ( 1 - T )^{9}( 1 + T )^{9} \)
31 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
37 \( ( 1 - T )^{9}( 1 + T )^{9} \)
41 \( ( 1 - T )^{9}( 1 + T )^{9} \)
43 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
47 \( ( 1 - T )^{9}( 1 + T )^{9} \)
53 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
59 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
61 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
67 \( ( 1 - T )^{9}( 1 + T )^{9} \)
71 \( ( 1 - T )^{9}( 1 + T )^{9} \)
73 \( ( 1 - T )^{9}( 1 + T )^{9} \)
79 \( ( 1 - T )^{9}( 1 + T )^{9} \)
83 \( ( 1 - T )^{9}( 1 + T )^{9} \)
89 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} \)
97 \( ( 1 - T )^{9}( 1 + T )^{9} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.44119621590577754522305425094, −3.35842119039145512656534998808, −3.35767279839851153005992813503, −3.18347522635212054020649452082, −3.13340014282033743856777523966, −2.87591866827579735819492635774, −2.68606942113178154329191009274, −2.62985274516886741972276610614, −2.62172026552413751554055710225, −2.50820128304426170633127938224, −2.48907637395639158098499668052, −2.07739599486409345876950567844, −2.06165720715763890207933870877, −2.04467322613614525190493885846, −1.78986516130767627794506068018, −1.71718114766405001909206501927, −1.67484527200720915243438708984, −1.53693588765050902039748055037, −1.21134958734021936250761052401, −1.11700597789192421439793534356, −1.01497794738284729244727139034, −0.991145402159650204754095168913, −0.987065152202113329429940796427, −0.851305936619547433527655763181, −0.809902879891093272743606034636, 0.809902879891093272743606034636, 0.851305936619547433527655763181, 0.987065152202113329429940796427, 0.991145402159650204754095168913, 1.01497794738284729244727139034, 1.11700597789192421439793534356, 1.21134958734021936250761052401, 1.53693588765050902039748055037, 1.67484527200720915243438708984, 1.71718114766405001909206501927, 1.78986516130767627794506068018, 2.04467322613614525190493885846, 2.06165720715763890207933870877, 2.07739599486409345876950567844, 2.48907637395639158098499668052, 2.50820128304426170633127938224, 2.62172026552413751554055710225, 2.62985274516886741972276610614, 2.68606942113178154329191009274, 2.87591866827579735819492635774, 3.13340014282033743856777523966, 3.18347522635212054020649452082, 3.35767279839851153005992813503, 3.35842119039145512656534998808, 3.44119621590577754522305425094

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

Plot not available for L-functions of degree greater than 10.