Properties

Label 4-758912-1.1-c1e2-0-43
Degree $4$
Conductor $758912$
Sign $-1$
Analytic cond. $48.3888$
Root an. cond. $2.63746$
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-s + 4-s + 8-s − 6·9-s − 2·11-s + 16-s + 4·17-s − 6·18-s − 2·22-s − 6·25-s + 32-s + 4·34-s − 6·36-s + 20·41-s + 8·43-s − 2·44-s + 49-s − 6·50-s + 64-s − 24·67-s + 4·68-s − 6·72-s − 28·73-s + 27·81-s + 20·82-s + 8·86-s − 2·88-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2·9-s − 0.603·11-s + 1/4·16-s + 0.970·17-s − 1.41·18-s − 0.426·22-s − 6/5·25-s + 0.176·32-s + 0.685·34-s − 36-s + 3.12·41-s + 1.21·43-s − 0.301·44-s + 1/7·49-s − 0.848·50-s + 1/8·64-s − 2.93·67-s + 0.485·68-s − 0.707·72-s − 3.27·73-s + 3·81-s + 2.20·82-s + 0.862·86-s − 0.213·88-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(758912\)    =    \(2^{7} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(48.3888\)
Root analytic conductor: \(2.63746\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{758912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 758912,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−7.72024110024881070821541339163, −7.58187479426412561326624344048, −7.52779608421941576346985997734, −6.42169131751856764457567127082, −6.02643591301064267650899026350, −5.69715804540238922198121355930, −5.60617960823028323278478834583, −4.82761578982425423372007838462, −4.30916127979318708251021123277, −3.78476385235783926950251020606, −3.08172507755908328232950357793, −2.72854508523448958533903853214, −2.30984363521644281586554494696, −1.21399850656019313589842252973, 0, 1.21399850656019313589842252973, 2.30984363521644281586554494696, 2.72854508523448958533903853214, 3.08172507755908328232950357793, 3.78476385235783926950251020606, 4.30916127979318708251021123277, 4.82761578982425423372007838462, 5.60617960823028323278478834583, 5.69715804540238922198121355930, 6.02643591301064267650899026350, 6.42169131751856764457567127082, 7.52779608421941576346985997734, 7.58187479426412561326624344048, 7.72024110024881070821541339163

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