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Properties

Label 4-48e4-1.1-c1e2-0-32
Degree $4$
Conductor $5308416$
Sign $1$
Analytic cond. $338.469$
Root an. cond. $4.28923$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·17-s − 6·25-s − 16·41-s − 14·49-s − 12·73-s + 32·89-s − 36·97-s − 32·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 3.88·17-s − 6/5·25-s − 2.49·41-s − 2·49-s − 1.40·73-s + 3.39·89-s − 3.65·97-s − 3.01·113-s + 2·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(338.469\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5308416,\ (\ :1/2, 1/2),\ 1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.17.q_du
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.41.q_fq
43$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.43.a_adi
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.a_adm
59$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.59.a_aeo
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.a_aw
67$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.67.a_afe
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.83.a_agk
89$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.89.abg_qs
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \) 2.97.bk_ty
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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

−8.775798061846830041730069472657, −8.582556288072288611105926068647, −7.959721195406703319186537047535, −7.896707631749010399782066658556, −6.98719002529485757407706228948, −6.89633917503043859512751270184, −6.58400705092942786927186466065, −6.20309480273145978676362427753, −5.72468607478491843059150078010, −5.13857742396956805203551427282, −4.71163983626368025729126972261, −4.47108203766416193137878529353, −4.00053368891146151824340777506, −3.49871481272708305499429267919, −2.94459232243783192743733298219, −2.26753885309882347217104592797, −2.00839927748616696443007480642, −1.45365171951982530450713265019, 0, 0, 1.45365171951982530450713265019, 2.00839927748616696443007480642, 2.26753885309882347217104592797, 2.94459232243783192743733298219, 3.49871481272708305499429267919, 4.00053368891146151824340777506, 4.47108203766416193137878529353, 4.71163983626368025729126972261, 5.13857742396956805203551427282, 5.72468607478491843059150078010, 6.20309480273145978676362427753, 6.58400705092942786927186466065, 6.89633917503043859512751270184, 6.98719002529485757407706228948, 7.896707631749010399782066658556, 7.959721195406703319186537047535, 8.582556288072288611105926068647, 8.775798061846830041730069472657

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