Properties

Label 4-877952-1.1-c1e2-0-1
Degree $4$
Conductor $877952$
Sign $-1$
Analytic cond. $55.9789$
Root an. cond. $2.73530$
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  − 4·3-s − 2·5-s + 6·9-s + 8·15-s + 10·17-s − 19-s − 7·25-s + 4·27-s + 16·31-s − 12·45-s − 5·49-s − 40·51-s + 4·57-s + 28·59-s − 10·61-s − 12·71-s − 30·73-s + 28·75-s − 8·79-s − 37·81-s − 20·85-s − 64·93-s + 2·95-s − 36·101-s − 28·103-s + 20·107-s − 13·121-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.894·5-s + 2·9-s + 2.06·15-s + 2.42·17-s − 0.229·19-s − 7/5·25-s + 0.769·27-s + 2.87·31-s − 1.78·45-s − 5/7·49-s − 5.60·51-s + 0.529·57-s + 3.64·59-s − 1.28·61-s − 1.42·71-s − 3.51·73-s + 3.23·75-s − 0.900·79-s − 4.11·81-s − 2.16·85-s − 6.63·93-s + 0.205·95-s − 3.58·101-s − 2.75·103-s + 1.93·107-s − 1.18·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 877952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 877952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(877952\)    =    \(2^{7} \cdot 19^{3}\)
Sign: $-1$
Analytic conductor: \(55.9789\)
Root analytic conductor: \(2.73530\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 877952,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
19$C_1$ \( 1 + T \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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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

−8.148060259363710086433322724847, −7.38601845396983265703380778176, −6.97775492717497319205239152750, −6.58341608756454848978175875806, −5.89152757780490661000300260672, −5.75508495262059909206416744964, −5.51991869338759908788687173901, −4.85172758929064856672456822692, −4.33693159410665846542938642673, −4.05153477392287557754730772390, −3.08915058635532111035563541614, −2.83555716155125599118268588753, −1.44011806260301581515132764673, −0.833864002051176659308197110715, 0, 0.833864002051176659308197110715, 1.44011806260301581515132764673, 2.83555716155125599118268588753, 3.08915058635532111035563541614, 4.05153477392287557754730772390, 4.33693159410665846542938642673, 4.85172758929064856672456822692, 5.51991869338759908788687173901, 5.75508495262059909206416744964, 5.89152757780490661000300260672, 6.58341608756454848978175875806, 6.97775492717497319205239152750, 7.38601845396983265703380778176, 8.148060259363710086433322724847

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