Properties

Label 4-1216e2-1.1-c0e2-0-0
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $0.368282$
Root an. cond. $0.779014$
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  − 2·9-s + 4·17-s + 2·25-s − 2·49-s − 4·73-s + 3·81-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-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 + ⋯
L(s)  = 1  − 2·9-s + 4·17-s + 2·25-s − 2·49-s − 4·73-s + 3·81-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.368282\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.009778879\)
\(L(\frac12)\) \(\approx\) \(1.009778879\)
\(L(1)\) 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 \( 1 \)
19$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 + T^{2} )^{2} \)
5$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$C_1$ \( ( 1 - T )^{4} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−10.03858683488733557500576328186, −9.788327927997988828700733405486, −9.336471726877595065706323718001, −8.846577655230746483293077608280, −8.332150890540909515735825309940, −8.318244500592858537973108643467, −7.60113094711622986676220564521, −7.51386320559291969345131698271, −6.83392596893603686582455254946, −6.25276101971310853271943309746, −5.74416293752405322785256604644, −5.69307302927373193768143545120, −5.06060394405552271932057231691, −4.84572381246513044076514900335, −3.90882605367159703739915161423, −3.27243948113723452794469111396, −2.98582072277298909962376996114, −2.86225146813457197621463621780, −1.62813312193582553836094851059, −0.973135725108569989388929430460, 0.973135725108569989388929430460, 1.62813312193582553836094851059, 2.86225146813457197621463621780, 2.98582072277298909962376996114, 3.27243948113723452794469111396, 3.90882605367159703739915161423, 4.84572381246513044076514900335, 5.06060394405552271932057231691, 5.69307302927373193768143545120, 5.74416293752405322785256604644, 6.25276101971310853271943309746, 6.83392596893603686582455254946, 7.51386320559291969345131698271, 7.60113094711622986676220564521, 8.318244500592858537973108643467, 8.332150890540909515735825309940, 8.846577655230746483293077608280, 9.336471726877595065706323718001, 9.788327927997988828700733405486, 10.03858683488733557500576328186

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