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Properties

Label 4-34e4-1.1-c0e2-0-0
Degree $4$
Conductor $1336336$
Sign $1$
Analytic cond. $0.332835$
Root an. cond. $0.759551$
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  − 2·2-s + 3·4-s − 4·8-s + 2·9-s + 5·16-s − 4·18-s − 6·32-s + 6·36-s + 2·49-s + 7·64-s − 8·72-s + 3·81-s − 4·98-s + 2·121-s + 127-s − 8·128-s + 131-s + 137-s + 139-s + 10·144-s + 149-s + 151-s + 157-s − 6·162-s + 163-s + 167-s − 2·169-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 2·9-s + 5·16-s − 4·18-s − 6·32-s + 6·36-s + 2·49-s + 7·64-s − 8·72-s + 3·81-s − 4·98-s + 2·121-s + 127-s − 8·128-s + 131-s + 137-s + 139-s + 10·144-s + 149-s + 151-s + 157-s − 6·162-s + 163-s + 167-s − 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1336336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1336336\)    =    \(2^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(0.332835\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1336336,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5285225440\)
\(L(\frac12)\) \(\approx\) \(0.5285225440\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
17 \( 1 \)
good3$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−10.07119591672476248371186517129, −9.947616488911611667805149713777, −9.253176209357900988401001777534, −9.151671926075898747539385697983, −8.616008782696301635635136454714, −8.220372922176685241258245346748, −7.66396180224138514176893932937, −7.44553303887263391257875235573, −7.02312437043636480573330298589, −6.83164465893778212385571895107, −6.11209684634065673106790697179, −5.95573941740973026735021173568, −5.14169447944175471979683650678, −4.63891897612742910562639083406, −3.79318585213712881284557812590, −3.61006690859387285815609488211, −2.61488359518915618297368300662, −2.23508563830816040785000016830, −1.49415910324576107748984374736, −1.01712437104286264984381406238, 1.01712437104286264984381406238, 1.49415910324576107748984374736, 2.23508563830816040785000016830, 2.61488359518915618297368300662, 3.61006690859387285815609488211, 3.79318585213712881284557812590, 4.63891897612742910562639083406, 5.14169447944175471979683650678, 5.95573941740973026735021173568, 6.11209684634065673106790697179, 6.83164465893778212385571895107, 7.02312437043636480573330298589, 7.44553303887263391257875235573, 7.66396180224138514176893932937, 8.220372922176685241258245346748, 8.616008782696301635635136454714, 9.151671926075898747539385697983, 9.253176209357900988401001777534, 9.947616488911611667805149713777, 10.07119591672476248371186517129

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