Properties

Label 4-112e3-1.1-c1e2-0-39
Degree $4$
Conductor $1404928$
Sign $1$
Analytic cond. $89.5794$
Root an. cond. $3.07646$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 6·9-s − 8·11-s − 6·25-s − 12·29-s + 4·37-s − 8·43-s + 49-s − 12·53-s − 6·63-s − 8·67-s + 16·71-s − 8·77-s − 32·79-s + 27·81-s + 48·99-s − 24·107-s + 20·109-s + 4·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 0.377·7-s − 2·9-s − 2.41·11-s − 6/5·25-s − 2.22·29-s + 0.657·37-s − 1.21·43-s + 1/7·49-s − 1.64·53-s − 0.755·63-s − 0.977·67-s + 1.89·71-s − 0.911·77-s − 3.60·79-s + 3·81-s + 4.82·99-s − 2.32·107-s + 1.91·109-s + 0.376·113-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1404928\)    =    \(2^{12} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(89.5794\)
Root analytic conductor: \(3.07646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1404928} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1404928,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7$C_1$ \( 1 - T \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.79292083571821938290976650763, −7.26382021666693896035142333427, −6.59264336057149769206055975154, −5.91740132798549001211232285063, −5.69394971603405508863657314834, −5.47493819124631815597461836959, −4.92225264716551837532790929786, −4.53211800578457486655010876487, −3.60173413237973468362473708205, −3.32773603072291404011890923635, −2.58182243115007516792385368988, −2.40145156924384381758797773878, −1.62407385510972851799092712087, 0, 0, 1.62407385510972851799092712087, 2.40145156924384381758797773878, 2.58182243115007516792385368988, 3.32773603072291404011890923635, 3.60173413237973468362473708205, 4.53211800578457486655010876487, 4.92225264716551837532790929786, 5.47493819124631815597461836959, 5.69394971603405508863657314834, 5.91740132798549001211232285063, 6.59264336057149769206055975154, 7.26382021666693896035142333427, 7.79292083571821938290976650763

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