Properties

Label 6-7942e3-1.1-c1e3-0-0
Degree $6$
Conductor $500944540888$
Sign $1$
Analytic cond. $255047.$
Root an. cond. $7.96349$
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  − 3·2-s − 3-s + 6·4-s + 5·5-s + 3·6-s + 7-s − 10·8-s − 3·9-s − 15·10-s − 3·11-s − 6·12-s − 5·13-s − 3·14-s − 5·15-s + 15·16-s + 4·17-s + 9·18-s + 30·20-s − 21-s + 9·22-s + 2·23-s + 10·24-s + 7·25-s + 15·26-s + 5·27-s + 6·28-s − 7·29-s + ⋯
L(s)  = 1  − 2.12·2-s − 0.577·3-s + 3·4-s + 2.23·5-s + 1.22·6-s + 0.377·7-s − 3.53·8-s − 9-s − 4.74·10-s − 0.904·11-s − 1.73·12-s − 1.38·13-s − 0.801·14-s − 1.29·15-s + 15/4·16-s + 0.970·17-s + 2.12·18-s + 6.70·20-s − 0.218·21-s + 1.91·22-s + 0.417·23-s + 2.04·24-s + 7/5·25-s + 2.94·26-s + 0.962·27-s + 1.13·28-s − 1.29·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 11^{3} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(255047.\)
Root analytic conductor: \(7.96349\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7942} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 11^{3} \cdot 19^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.251177495\)
\(L(\frac12)\) \(\approx\) \(1.251177495\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{3} \)
11$C_1$ \( ( 1 + T )^{3} \)
19 \( 1 \)
good3$S_4\times C_2$ \( 1 + T + 4 T^{2} + 2 T^{3} + 4 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 - p T + 18 T^{2} - 48 T^{3} + 18 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - T + 2 p T^{2} - 18 T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 5 T + 40 T^{2} + 116 T^{3} + 40 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 4 T + 27 T^{2} - 48 T^{3} + 27 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 2 T + 41 T^{2} - 124 T^{3} + 41 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 7 T + 68 T^{2} + 320 T^{3} + 68 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 3 T + 68 T^{2} - 110 T^{3} + 68 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 14 T + 127 T^{2} - 908 T^{3} + 127 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 3 T + 98 T^{2} - 268 T^{3} + 98 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 5 T + 62 T^{2} + 162 T^{3} + 62 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 89 T^{2} + 128 T^{3} + 89 p T^{4} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 16 T + 223 T^{2} - 1752 T^{3} + 223 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 12 T + 17 T^{2} + 376 T^{3} + 17 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
67$S_4\times C_2$ \( 1 + T + 194 T^{2} + 138 T^{3} + 194 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 3 T + 182 T^{2} - 454 T^{3} + 182 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 4 T + 83 T^{2} - 528 T^{3} + 83 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 2 T + 121 T^{2} + 668 T^{3} + 121 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 7 T + 106 T^{2} - 54 T^{3} + 106 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 16 T - 13 T^{2} - 1576 T^{3} - 13 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 14 T + 307 T^{2} - 2588 T^{3} + 307 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.09252305861686730828329194085, −6.54665106083361868285376464920, −6.52571432279369548904526881768, −6.42390929806197203726979148015, −5.90300803475461354973839020729, −5.77109712612254622740352961694, −5.68669838526825236550310536567, −5.36085089733899590334804972438, −5.25250027381262449544238711551, −5.21446528244165584270630327272, −4.50763448085514694389951685713, −4.49174878561605892290820500977, −4.11711298645048061117821976669, −3.41849510289572149103247860793, −3.36751253372305631771039999052, −3.07269041421224312029660906665, −2.59874359347062607742585882298, −2.42547993265815295911299553356, −2.33269587707357491858031002617, −2.06195749525812658896100631395, −1.75155092248507693176569239097, −1.40826110029927433936355435417, −0.955507814206067275387571053669, −0.64118461406034651318597407850, −0.33407510291370773399809401488, 0.33407510291370773399809401488, 0.64118461406034651318597407850, 0.955507814206067275387571053669, 1.40826110029927433936355435417, 1.75155092248507693176569239097, 2.06195749525812658896100631395, 2.33269587707357491858031002617, 2.42547993265815295911299553356, 2.59874359347062607742585882298, 3.07269041421224312029660906665, 3.36751253372305631771039999052, 3.41849510289572149103247860793, 4.11711298645048061117821976669, 4.49174878561605892290820500977, 4.50763448085514694389951685713, 5.21446528244165584270630327272, 5.25250027381262449544238711551, 5.36085089733899590334804972438, 5.68669838526825236550310536567, 5.77109712612254622740352961694, 5.90300803475461354973839020729, 6.42390929806197203726979148015, 6.52571432279369548904526881768, 6.54665106083361868285376464920, 7.09252305861686730828329194085

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