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Properties

Label 4-739328-1.1-c1e2-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·9-s + 6·13-s + 6·17-s + 10·25-s − 2·29-s + 12·37-s − 5·49-s − 2·53-s + 6·73-s + 16·81-s − 18·109-s − 30·117-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 30·153-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 5/3·9-s + 1.66·13-s + 1.45·17-s + 2·25-s − 0.371·29-s + 1.97·37-s − 5/7·49-s − 0.274·53-s + 0.702·73-s + 16/9·81-s − 1.72·109-s − 2.77·117-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 − 2.42·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 739328,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.039381637\)
\(L(\frac12)\) \(\approx\) \(2.039381637\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
19$C_2$ \( 1 + p T^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.13.ag_bj
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.23.a_bt
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.29.c_ch
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.41.a_aco
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.a_da
53$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.53.c_ed
59$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.a_bl
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.67.a_ev
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.73.ag_fz
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.79.a_o
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.83.a_dy
89$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.89.a_aek
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.97.a_aby
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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

−8.198756302579373184438464485610, −8.033711221863498163412112505556, −7.56105964780757381977062698937, −6.80772265126106879091598745037, −6.42746415180518078389640033322, −6.01455507612461530329647528693, −5.59324354159366978587093356376, −5.23788503197695929628445168517, −4.62235309372838785711525884113, −3.94244083248322658277381436869, −3.35711507579402047318520084256, −3.02316528850583213020940304971, −2.49331505172094797326240869789, −1.41167529829377980851763217011, −0.76692217829493386761159836301, 0.76692217829493386761159836301, 1.41167529829377980851763217011, 2.49331505172094797326240869789, 3.02316528850583213020940304971, 3.35711507579402047318520084256, 3.94244083248322658277381436869, 4.62235309372838785711525884113, 5.23788503197695929628445168517, 5.59324354159366978587093356376, 6.01455507612461530329647528693, 6.42746415180518078389640033322, 6.80772265126106879091598745037, 7.56105964780757381977062698937, 8.033711221863498163412112505556, 8.198756302579373184438464485610

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