Properties

Label 4-442368-1.1-c1e2-0-74
Degree $4$
Conductor $442368$
Sign $1$
Analytic cond. $28.2057$
Root an. cond. $2.30454$
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  + 3-s − 8·5-s + 9-s − 8·15-s − 16·19-s − 8·23-s + 38·25-s + 27-s − 8·45-s − 8·47-s − 10·49-s − 16·57-s − 8·67-s − 8·69-s − 24·71-s − 20·73-s + 38·75-s + 81-s + 128·95-s + 20·97-s + 64·115-s − 6·121-s − 136·125-s + 127-s + 131-s − 8·135-s + 137-s + ⋯
L(s)  = 1  + 0.577·3-s − 3.57·5-s + 1/3·9-s − 2.06·15-s − 3.67·19-s − 1.66·23-s + 38/5·25-s + 0.192·27-s − 1.19·45-s − 1.16·47-s − 1.42·49-s − 2.11·57-s − 0.977·67-s − 0.963·69-s − 2.84·71-s − 2.34·73-s + 4.38·75-s + 1/9·81-s + 13.1·95-s + 2.03·97-s + 5.96·115-s − 0.545·121-s − 12.1·125-s + 0.0887·127-s + 0.0873·131-s − 0.688·135-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(442368\)    =    \(2^{14} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(28.2057\)
Root analytic conductor: \(2.30454\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 442368,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−8.230854059399453444196153124887, −7.80892757070983972668727076655, −7.30623361332829545195635303299, −7.08289458029334266024179255426, −6.27798985919626463740519529757, −6.13258408514753038234534207414, −4.69737181089599911757008902155, −4.56007962912554840821971374527, −4.20448032364885463054764539919, −3.77214318657172177768570145308, −3.32102035102375871114821653214, −2.65190583901467307643981603895, −1.70795903707382777058669654679, 0, 0, 1.70795903707382777058669654679, 2.65190583901467307643981603895, 3.32102035102375871114821653214, 3.77214318657172177768570145308, 4.20448032364885463054764539919, 4.56007962912554840821971374527, 4.69737181089599911757008902155, 6.13258408514753038234534207414, 6.27798985919626463740519529757, 7.08289458029334266024179255426, 7.30623361332829545195635303299, 7.80892757070983972668727076655, 8.230854059399453444196153124887

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