Properties

Label 4-739328-1.1-c1e2-0-7
Degree $4$
Conductor $739328$
Sign $-1$
Analytic cond. $47.1401$
Root an. cond. $2.62028$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5·9-s + 10·17-s + 6·25-s − 24·41-s − 13·49-s − 14·73-s + 16·81-s − 12·89-s + 12·97-s − 12·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 50·153-s + 157-s + 163-s + 167-s + 25·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 5/3·9-s + 2.42·17-s + 6/5·25-s − 3.74·41-s − 1.85·49-s − 1.63·73-s + 16/9·81-s − 1.27·89-s + 1.21·97-s − 1.12·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 4.04·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 739328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 739328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(739328\)    =    \(2^{11} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(47.1401\)
Root analytic conductor: \(2.62028\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{739328} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 739328,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.097977130148826306537543729545, −7.78270604734653611245586904422, −7.17256961441030457549618001314, −6.66497413006612600364626607654, −6.30023261682060838291785304978, −5.64331877998247196033481429906, −5.35462021395561592211452786575, −5.05471255286984233114047950550, −4.38790175941963038428793452679, −3.39826650555000626660333262631, −3.25408706841441563001400294407, −2.91676218961444217862223110393, −1.89564027396146231703355154113, −1.17643257441768604939588188061, 0, 1.17643257441768604939588188061, 1.89564027396146231703355154113, 2.91676218961444217862223110393, 3.25408706841441563001400294407, 3.39826650555000626660333262631, 4.38790175941963038428793452679, 5.05471255286984233114047950550, 5.35462021395561592211452786575, 5.64331877998247196033481429906, 6.30023261682060838291785304978, 6.66497413006612600364626607654, 7.17256961441030457549618001314, 7.78270604734653611245586904422, 8.097977130148826306537543729545

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