Properties

Label 4-671e2-1.1-c1e2-0-1
Degree $4$
Conductor $450241$
Sign $-1$
Analytic cond. $28.7077$
Root an. cond. $2.31472$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s − 3·9-s + 8·13-s − 4·15-s − 4·16-s − 7·25-s + 14·27-s − 16·39-s − 16·41-s − 6·45-s + 16·47-s + 8·48-s − 10·49-s + 12·61-s + 16·65-s + 8·73-s + 14·75-s − 8·80-s − 4·81-s − 12·83-s − 14·97-s − 32·103-s + 36·107-s + 20·109-s + 18·113-s − 24·117-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 9-s + 2.21·13-s − 1.03·15-s − 16-s − 7/5·25-s + 2.69·27-s − 2.56·39-s − 2.49·41-s − 0.894·45-s + 2.33·47-s + 1.15·48-s − 1.42·49-s + 1.53·61-s + 1.98·65-s + 0.936·73-s + 1.61·75-s − 0.894·80-s − 4/9·81-s − 1.31·83-s − 1.42·97-s − 3.15·103-s + 3.48·107-s + 1.91·109-s + 1.69·113-s − 2.21·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450241 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(450241\)    =    \(11^{2} \cdot 61^{2}\)
Sign: $-1$
Analytic conductor: \(28.7077\)
Root analytic conductor: \(2.31472\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 450241,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
61$C_2$ \( 1 - 12 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−8.603539619290756001226038948684, −8.081040062265620066406135132289, −7.28318335450384603202768539700, −6.67660370674787540556278099536, −6.36261389471308870138602900888, −5.93503465259105862079796232003, −5.72745892885950030255248689782, −5.21215205106375072423142352585, −4.73510070440563314601051738956, −3.84660431078072848220887031878, −3.52054438791519721780926007468, −2.65326845947239220636806267917, −2.00570631269277014788928500184, −1.16014008983661363091540869692, 0, 1.16014008983661363091540869692, 2.00570631269277014788928500184, 2.65326845947239220636806267917, 3.52054438791519721780926007468, 3.84660431078072848220887031878, 4.73510070440563314601051738956, 5.21215205106375072423142352585, 5.72745892885950030255248689782, 5.93503465259105862079796232003, 6.36261389471308870138602900888, 6.67660370674787540556278099536, 7.28318335450384603202768539700, 8.081040062265620066406135132289, 8.603539619290756001226038948684

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