Properties

Degree 6
Conductor $ 2003^{3} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s + 6·16-s − 3·19-s + 3·25-s − 27-s + 3·49-s − 3·53-s + 10·64-s − 9·76-s − 3·89-s + 9·100-s − 3·108-s − 3·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 3·4-s + 6·16-s − 3·19-s + 3·25-s − 27-s + 3·49-s − 3·53-s + 10·64-s − 9·76-s − 3·89-s + 9·100-s − 3·108-s − 3·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2003^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2003^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(2003^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{2003} (2002, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(6,\ 2003^{3} ,\ ( \ : 0, 0, 0 ),\ 1 )$
$L(\frac{1}{2})$  $\approx$  $3.036622354$
$L(\frac12)$  $\approx$  $3.036622354$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2003$, \(F_p\) is a polynomial of degree 6. If $p = 2003$, then $F_p$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p$
bad2003 \( 1+O(T) \)
good2$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
3$C_6$ \( 1 + T^{3} + T^{6} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
13$C_6$ \( 1 + T^{3} + T^{6} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
19$C_2$ \( ( 1 + T + T^{2} )^{3} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
47$C_6$ \( 1 + T^{3} + T^{6} \)
53$C_2$ \( ( 1 + T + T^{2} )^{3} \)
59$C_6$ \( 1 + T^{3} + T^{6} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
73$C_6$ \( 1 + T^{3} + T^{6} \)
79$C_6$ \( 1 + T^{3} + T^{6} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
89$C_2$ \( ( 1 + T + T^{2} )^{3} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.428208883916818202770840587223, −7.85305608860504719413333730021, −7.84498276518176958371490651454, −7.41712167701710131552400132536, −7.12960199598308512493803321698, −6.95700635433192104564140022369, −6.74031035280392642966854844786, −6.54400559867713352674993768852, −6.24649343785626698549659674428, −6.13648691027400823583051673066, −5.68031559858507047814753067409, −5.52519585731075404240487209219, −5.25384165983110084796513466485, −4.71935729270348960319489264639, −4.30350435254199879302278573543, −4.27648195618746220461459441030, −3.69986029787874061932208308603, −3.30057283543544462659792325523, −3.14331586431822676243864670141, −2.58262598939111800571373723922, −2.55635132839001603736068856688, −2.21453177929666434446917857794, −1.80124379298909506691068598074, −1.43042549484316165441871962923, −1.03009713626109061612792543694, 1.03009713626109061612792543694, 1.43042549484316165441871962923, 1.80124379298909506691068598074, 2.21453177929666434446917857794, 2.55635132839001603736068856688, 2.58262598939111800571373723922, 3.14331586431822676243864670141, 3.30057283543544462659792325523, 3.69986029787874061932208308603, 4.27648195618746220461459441030, 4.30350435254199879302278573543, 4.71935729270348960319489264639, 5.25384165983110084796513466485, 5.52519585731075404240487209219, 5.68031559858507047814753067409, 6.13648691027400823583051673066, 6.24649343785626698549659674428, 6.54400559867713352674993768852, 6.74031035280392642966854844786, 6.95700635433192104564140022369, 7.12960199598308512493803321698, 7.41712167701710131552400132536, 7.84498276518176958371490651454, 7.85305608860504719413333730021, 8.428208883916818202770840587223

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