Properties

Label 6-7800e3-1.1-c1e3-0-12
Degree $6$
Conductor $474552000000$
Sign $-1$
Analytic cond. $241610.$
Root an. cond. $7.89197$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 7-s + 6·9-s − 3·11-s − 3·13-s − 3·17-s − 6·19-s + 3·21-s + 10·27-s − 29-s − 7·31-s − 9·33-s − 14·37-s − 9·39-s + 4·43-s + 47-s − 13·49-s − 9·51-s − 5·53-s − 18·57-s − 7·59-s − 5·61-s + 6·63-s − 3·67-s − 10·71-s + 10·73-s − 3·77-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.377·7-s + 2·9-s − 0.904·11-s − 0.832·13-s − 0.727·17-s − 1.37·19-s + 0.654·21-s + 1.92·27-s − 0.185·29-s − 1.25·31-s − 1.56·33-s − 2.30·37-s − 1.44·39-s + 0.609·43-s + 0.145·47-s − 1.85·49-s − 1.26·51-s − 0.686·53-s − 2.38·57-s − 0.911·59-s − 0.640·61-s + 0.755·63-s − 0.366·67-s − 1.18·71-s + 1.17·73-s − 0.341·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\)
Sign: $-1$
Analytic conductor: \(241610.\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
5 \( 1 \)
13$C_1$ \( ( 1 + T )^{3} \)
good7$S_4\times C_2$ \( 1 - T + 2 p T^{2} - 17 T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 3 T + 16 T^{2} + 13 T^{3} + 16 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
17$C_2$ \( ( 1 + T + p T^{2} )^{3} \)
19$S_4\times C_2$ \( 1 + 6 T + 37 T^{2} + 128 T^{3} + 37 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 37 T^{2} + 44 T^{3} + 37 p T^{4} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + T + 34 T^{2} - 35 T^{3} + 34 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 7 T + 28 T^{2} + 9 T^{3} + 28 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 14 T + 167 T^{2} + 1096 T^{3} + 167 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 39 T^{2} - 52 T^{3} + 39 p T^{4} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 4 T + 125 T^{2} - 332 T^{3} + 125 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - T + 58 T^{2} - 5 T^{3} + 58 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 5 T + 34 T^{2} - 251 T^{3} + 34 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 7 T + 142 T^{2} + 835 T^{3} + 142 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 5 T + 138 T^{2} + 601 T^{3} + 138 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 3 T + 184 T^{2} + 417 T^{3} + 184 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 10 T + 201 T^{2} + 1240 T^{3} + 201 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 10 T + 159 T^{2} - 1512 T^{3} + 159 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 16 T + 313 T^{2} + 2628 T^{3} + 313 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 11 T + 238 T^{2} + 1571 T^{3} + 238 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 8 T + 235 T^{2} + 1376 T^{3} + 235 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 26 T + 487 T^{2} + 5484 T^{3} + 487 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.36645298664567748155529582293, −7.17673928439489633514636083625, −6.89454872223382602174585629331, −6.74826729799244326616851944656, −6.34684616556637364500724730651, −6.26822328567150859116702055305, −5.97566201735451679822998892963, −5.47172987119443798113192483332, −5.28768935166878612199687951321, −5.27231316095327368916784673723, −4.77176406187070853417758546222, −4.64394502821784250997703741371, −4.48460188821038210185869096401, −3.96455204586088306593940314204, −3.88802356237877723893602003907, −3.81650951315804025123480549992, −3.22527335795665915036373892332, −3.08393306460702167811867346805, −2.75294442567611409536773755567, −2.61398239626680768403055755465, −2.23759444606215977270569269316, −2.11311745575661065775572778395, −1.58157505950349790360055074143, −1.41888954097044174978271753810, −1.33759323868721965817113338273, 0, 0, 0, 1.33759323868721965817113338273, 1.41888954097044174978271753810, 1.58157505950349790360055074143, 2.11311745575661065775572778395, 2.23759444606215977270569269316, 2.61398239626680768403055755465, 2.75294442567611409536773755567, 3.08393306460702167811867346805, 3.22527335795665915036373892332, 3.81650951315804025123480549992, 3.88802356237877723893602003907, 3.96455204586088306593940314204, 4.48460188821038210185869096401, 4.64394502821784250997703741371, 4.77176406187070853417758546222, 5.27231316095327368916784673723, 5.28768935166878612199687951321, 5.47172987119443798113192483332, 5.97566201735451679822998892963, 6.26822328567150859116702055305, 6.34684616556637364500724730651, 6.74826729799244326616851944656, 6.89454872223382602174585629331, 7.17673928439489633514636083625, 7.36645298664567748155529582293

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