Properties

Label 4-2352e2-1.1-c1e2-0-3
Degree $4$
Conductor $5531904$
Sign $1$
Analytic cond. $352.718$
Root an. cond. $4.33368$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 4·11-s + 4·13-s + 2·15-s − 6·17-s − 4·19-s + 5·25-s + 27-s − 4·29-s − 4·33-s − 6·37-s − 4·39-s − 4·41-s + 8·43-s + 6·51-s − 6·53-s − 8·55-s + 4·57-s − 12·59-s − 2·61-s − 8·65-s + 4·67-s − 6·73-s − 5·75-s − 16·79-s − 81-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1.20·11-s + 1.10·13-s + 0.516·15-s − 1.45·17-s − 0.917·19-s + 25-s + 0.192·27-s − 0.742·29-s − 0.696·33-s − 0.986·37-s − 0.640·39-s − 0.624·41-s + 1.21·43-s + 0.840·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s − 1.56·59-s − 0.256·61-s − 0.992·65-s + 0.488·67-s − 0.702·73-s − 0.577·75-s − 1.80·79-s − 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(352.718\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5531904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4651545348\)
\(L(\frac12)\) \(\approx\) \(0.4651545348\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.300118321211605063923040790306, −8.752076864244550689075817986574, −8.385784379800755807502046996465, −8.256156235302716107394549011237, −7.56402726499231599570365145860, −7.09330532901617981437603416994, −6.87045450635048356605933690787, −6.49359849011394730112873966285, −6.16181959119288311607727680876, −5.64810322739201402753811284843, −5.37447014127236160357829629877, −4.44575502710521987012213463275, −4.43352542860517706390067571029, −4.08793253634363500271815482629, −3.55719680809943734518882248158, −3.03725136147849304515119055929, −2.50372792709442736496260014304, −1.51572481975669967895345570004, −1.44706056803683941569737165584, −0.24886185022786222439582601598, 0.24886185022786222439582601598, 1.44706056803683941569737165584, 1.51572481975669967895345570004, 2.50372792709442736496260014304, 3.03725136147849304515119055929, 3.55719680809943734518882248158, 4.08793253634363500271815482629, 4.43352542860517706390067571029, 4.44575502710521987012213463275, 5.37447014127236160357829629877, 5.64810322739201402753811284843, 6.16181959119288311607727680876, 6.49359849011394730112873966285, 6.87045450635048356605933690787, 7.09330532901617981437603416994, 7.56402726499231599570365145860, 8.256156235302716107394549011237, 8.385784379800755807502046996465, 8.752076864244550689075817986574, 9.300118321211605063923040790306

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