Properties

Label 4-800384-1.1-c1e2-0-0
Degree $4$
Conductor $800384$
Sign $-1$
Analytic cond. $51.0331$
Root an. cond. $2.67277$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 4·9-s − 2·13-s + 7·17-s − 3·25-s − 6·29-s + 2·37-s + 9·41-s + 12·45-s + 6·49-s + 53-s − 6·61-s + 6·65-s + 27·73-s + 7·81-s − 21·85-s + 3·89-s − 18·97-s − 13·101-s − 21·109-s + 39·113-s + 8·117-s + 15·121-s + 30·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.34·5-s − 4/3·9-s − 0.554·13-s + 1.69·17-s − 3/5·25-s − 1.11·29-s + 0.328·37-s + 1.40·41-s + 1.78·45-s + 6/7·49-s + 0.137·53-s − 0.768·61-s + 0.744·65-s + 3.16·73-s + 7/9·81-s − 2.27·85-s + 0.317·89-s − 1.82·97-s − 1.29·101-s − 2.01·109-s + 3.66·113-s + 0.739·117-s + 1.36·121-s + 2.68·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(800384\)    =    \(2^{7} \cdot 13^{2} \cdot 37\)
Sign: $-1$
Analytic conductor: \(51.0331\)
Root analytic conductor: \(2.67277\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 800384,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 3 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \)
19$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 51 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \)
59$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 11 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 129 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−7.917939833095993676533297660225, −7.65010591666647941866464963891, −7.40053768627103976407539242648, −6.79035167899595959795658746889, −6.07445796535013780032241986773, −5.69560299467617460234089983612, −5.44206114614174896713964280730, −4.77985994314594267077533489697, −4.15778174628688710730491450011, −3.68623384396250968546785531482, −3.34454043013029617320640300353, −2.68328570602431600063470499629, −2.07462866304559559006807273816, −0.902507847438035719106670432727, 0, 0.902507847438035719106670432727, 2.07462866304559559006807273816, 2.68328570602431600063470499629, 3.34454043013029617320640300353, 3.68623384396250968546785531482, 4.15778174628688710730491450011, 4.77985994314594267077533489697, 5.44206114614174896713964280730, 5.69560299467617460234089983612, 6.07445796535013780032241986773, 6.79035167899595959795658746889, 7.40053768627103976407539242648, 7.65010591666647941866464963891, 7.917939833095993676533297660225

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