Properties

Degree $4$
Conductor $415872$
Sign $-1$
Motivic weight $1$
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 − 4·5-s + 3·9-s − 8·15-s + 4·17-s + 4·19-s + 2·25-s + 4·27-s − 16·31-s − 12·45-s − 14·49-s + 8·51-s + 8·57-s − 8·59-s − 4·61-s + 8·67-s − 16·71-s + 20·73-s + 4·75-s + 16·79-s + 5·81-s − 16·85-s − 32·93-s − 16·95-s − 36·101-s − 32·103-s + 24·107-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s + 9-s − 2.06·15-s + 0.970·17-s + 0.917·19-s + 2/5·25-s + 0.769·27-s − 2.87·31-s − 1.78·45-s − 2·49-s + 1.12·51-s + 1.05·57-s − 1.04·59-s − 0.512·61-s + 0.977·67-s − 1.89·71-s + 2.34·73-s + 0.461·75-s + 1.80·79-s + 5/9·81-s − 1.73·85-s − 3.31·93-s − 1.64·95-s − 3.58·101-s − 3.15·103-s + 2.32·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(415872\)    =    \(2^{7} \cdot 3^{2} \cdot 19^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{415872} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 415872,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 - T )^{2} \)
19$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.173282349834764667604075089133, −7.979156449589182873646432658217, −7.61950951666595077126724317917, −7.26958585488936859387169650645, −6.85891827919805546068039331531, −6.09791694801381232421198374318, −5.40997773154525098146238432632, −5.01357758335939619070346679818, −4.23309352016035067675133294948, −3.83764998170958613523904255647, −3.37632188765103348554746581671, −3.14826068230388029805668446658, −2.12120802782336302555457408398, −1.36402928363394079803758664889, 0, 1.36402928363394079803758664889, 2.12120802782336302555457408398, 3.14826068230388029805668446658, 3.37632188765103348554746581671, 3.83764998170958613523904255647, 4.23309352016035067675133294948, 5.01357758335939619070346679818, 5.40997773154525098146238432632, 6.09791694801381232421198374318, 6.85891827919805546068039331531, 7.26958585488936859387169650645, 7.61950951666595077126724317917, 7.979156449589182873646432658217, 8.173282349834764667604075089133

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