Properties

Label 6-3696e3-1.1-c1e3-0-1
Degree $6$
Conductor $50488897536$
Sign $1$
Analytic cond. $25705.5$
Root an. cond. $5.43256$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 4·5-s + 3·7-s + 6·9-s + 3·11-s − 4·13-s − 12·15-s + 8·17-s + 8·19-s − 9·21-s − 10·23-s + 8·25-s − 10·27-s − 4·29-s + 2·31-s − 9·33-s + 12·35-s + 12·39-s + 14·41-s + 14·43-s + 24·45-s + 6·49-s − 24·51-s + 12·55-s − 24·57-s − 6·61-s + 18·63-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.78·5-s + 1.13·7-s + 2·9-s + 0.904·11-s − 1.10·13-s − 3.09·15-s + 1.94·17-s + 1.83·19-s − 1.96·21-s − 2.08·23-s + 8/5·25-s − 1.92·27-s − 0.742·29-s + 0.359·31-s − 1.56·33-s + 2.02·35-s + 1.92·39-s + 2.18·41-s + 2.13·43-s + 3.57·45-s + 6/7·49-s − 3.36·51-s + 1.61·55-s − 3.17·57-s − 0.768·61-s + 2.26·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(25705.5\)
Root analytic conductor: \(5.43256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3696} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 11^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.359725681\)
\(L(\frac12)\) \(\approx\) \(5.359725681\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{3} \)
7$C_1$ \( ( 1 - T )^{3} \)
11$C_1$ \( ( 1 - T )^{3} \)
good5$S_4\times C_2$ \( 1 - 4 T + 8 T^{2} - 14 T^{3} + 8 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 4 T + 12 T^{2} + 10 T^{3} + 12 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 8 T + 11 T^{2} + 56 T^{3} + 11 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 8 T + 72 T^{2} - 308 T^{3} + 72 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 10 T + 81 T^{2} + 396 T^{3} + 81 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 4 T + 60 T^{2} + 138 T^{3} + 60 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 2 T + 17 T^{2} + 132 T^{3} + 17 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 68 T^{2} + 106 T^{3} + 68 p T^{4} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 14 T + 163 T^{2} - 1116 T^{3} + 163 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 14 T + 85 T^{2} - 356 T^{3} + 85 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 80 T^{2} - 32 T^{3} + 80 p T^{4} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 143 T^{2} + 8 T^{3} + 143 p T^{4} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 120 T^{2} + 52 T^{3} + 120 p T^{4} + p^{3} T^{6} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{3} \)
67$S_4\times C_2$ \( 1 - 4 T + 116 T^{2} - 300 T^{3} + 116 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 12 T + 197 T^{2} - 1448 T^{3} + 197 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 20 T + 320 T^{2} + 3054 T^{3} + 320 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 12 T + 221 T^{2} + 1640 T^{3} + 221 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 6 T + 117 T^{2} + 500 T^{3} + 117 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 26 T + 407 T^{2} - 4300 T^{3} + 407 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 4 T + 171 T^{2} + 544 T^{3} + 171 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.49788177438515530215528087373, −7.29532204584028897990879339315, −7.09734673673948333306587917000, −6.83660427491770694418893151981, −6.27705042480643582005394971925, −6.01932050886035694863227706430, −6.01486432050076410452735179307, −5.90135329968953857902734340282, −5.52194199390554677415861790925, −5.40726541664698431168493589470, −5.15921322296804708298016368780, −4.77399972498254181123663549150, −4.70596156251043827502147211548, −4.12751797562809132546973901229, −3.97929745791476547864284670174, −3.94787196484250245389805423671, −3.17843405199396734062401179600, −2.86061801858271483756255677168, −2.75868645761216958515863969917, −1.89182752987041465621278356610, −1.88171481368921067785699520471, −1.84120696627217694691777653612, −1.03220033152491000292246679950, −0.964760919634283298860973572278, −0.57475897874472152869261985363, 0.57475897874472152869261985363, 0.964760919634283298860973572278, 1.03220033152491000292246679950, 1.84120696627217694691777653612, 1.88171481368921067785699520471, 1.89182752987041465621278356610, 2.75868645761216958515863969917, 2.86061801858271483756255677168, 3.17843405199396734062401179600, 3.94787196484250245389805423671, 3.97929745791476547864284670174, 4.12751797562809132546973901229, 4.70596156251043827502147211548, 4.77399972498254181123663549150, 5.15921322296804708298016368780, 5.40726541664698431168493589470, 5.52194199390554677415861790925, 5.90135329968953857902734340282, 6.01486432050076410452735179307, 6.01932050886035694863227706430, 6.27705042480643582005394971925, 6.83660427491770694418893151981, 7.09734673673948333306587917000, 7.29532204584028897990879339315, 7.49788177438515530215528087373

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