Properties

Label 4-768e2-1.1-c0e2-0-0
Degree $4$
Conductor $589824$
Sign $1$
Analytic cond. $0.146905$
Root an. cond. $0.619097$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 9-s + 2·25-s − 2·49-s + 4·73-s + 81-s + 4·97-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 2·225-s + ⋯
L(s)  = 1  − 9-s + 2·25-s − 2·49-s + 4·73-s + 81-s + 4·97-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 2·225-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(589824\)    =    \(2^{16} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(0.146905\)
Root analytic conductor: \(0.619097\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{768} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 589824,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8249252067\)
\(L(\frac12)\) \(\approx\) \(0.8249252067\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
good5$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$ \( ( 1 - T )^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$ \( ( 1 - T )^{4} \)
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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

−10.58411748633036289322809683421, −10.57579604182470278989178742714, −9.842610997450187228092650115146, −9.309281712207652705462010343223, −9.183908354581915427311547196559, −8.503231277316214331094544444154, −8.296824494016569845828844685206, −7.83229911155073064036663204386, −7.29631093228283246496110921738, −6.74423643649784553539034866123, −6.35265223941422036836609645592, −6.04541178656737113000583711432, −5.18517096887695521932682548074, −5.04220861762749904743178176906, −4.56634549720434133804464441339, −3.56565119294518014715373598869, −3.41007626517704594129377904426, −2.63984763760398080177827851398, −2.11411251405953769947321261844, −1.02777572148403383408704263892, 1.02777572148403383408704263892, 2.11411251405953769947321261844, 2.63984763760398080177827851398, 3.41007626517704594129377904426, 3.56565119294518014715373598869, 4.56634549720434133804464441339, 5.04220861762749904743178176906, 5.18517096887695521932682548074, 6.04541178656737113000583711432, 6.35265223941422036836609645592, 6.74423643649784553539034866123, 7.29631093228283246496110921738, 7.83229911155073064036663204386, 8.296824494016569845828844685206, 8.503231277316214331094544444154, 9.183908354581915427311547196559, 9.309281712207652705462010343223, 9.842610997450187228092650115146, 10.57579604182470278989178742714, 10.58411748633036289322809683421

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