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Properties

Label 4-3328e2-1.1-c1e2-0-2
Degree $4$
Conductor $11075584$
Sign $1$
Analytic cond. $706.188$
Root an. cond. $5.15501$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 6·9-s − 2·13-s + 8·17-s + 2·25-s − 4·29-s − 12·37-s + 4·41-s + 24·45-s + 4·49-s − 16·53-s + 8·65-s + 12·73-s + 27·81-s − 32·85-s + 20·89-s + 36·97-s − 8·101-s + 4·109-s + 32·113-s + 12·117-s − 4·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1.78·5-s − 2·9-s − 0.554·13-s + 1.94·17-s + 2/5·25-s − 0.742·29-s − 1.97·37-s + 0.624·41-s + 3.57·45-s + 4/7·49-s − 2.19·53-s + 0.992·65-s + 1.40·73-s + 3·81-s − 3.47·85-s + 2.11·89-s + 3.65·97-s − 0.796·101-s + 0.383·109-s + 3.01·113-s + 1.10·117-s − 0.363·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11075584\)    =    \(2^{16} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(706.188\)
Root analytic conductor: \(5.15501\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11075584,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4927396255\)
\(L(\frac12)\) \(\approx\) \(0.4927396255\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.7.a_ae
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.11.a_e
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.17.ai_by
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.19.a_u
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.31.a_bs
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \) 2.47.a_acq
53$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.53.q_go
59$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \) 2.59.a_dw
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \) 2.67.a_em
71$C_2^2$ \( 1 + 124 T^{2} + p^{2} T^{4} \) 2.71.a_eu
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.79.a_di
83$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.83.a_e
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \) 2.97.abk_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

−8.556702936501436208141737692509, −8.541485438164292735418107891576, −7.84452220382652342846946528697, −7.81995041110292028237710070100, −7.46650868845705331512986323841, −7.27997338241203579758958449557, −6.37297320641009731219027408655, −6.29418978351181303233847587232, −5.71171212591502287184577448974, −5.43832227181315514769757423559, −4.84147234427561243175964105754, −4.83939395504323857616301277266, −3.91496065289375490933073148483, −3.63375912994371950455679240281, −3.21330476887886852338836372284, −3.21227472932081915437404400156, −2.30458694242691921447285508621, −1.90765950066416831346590094818, −0.876470278411069950884027055131, −0.27374792528774296954263065223, 0.27374792528774296954263065223, 0.876470278411069950884027055131, 1.90765950066416831346590094818, 2.30458694242691921447285508621, 3.21227472932081915437404400156, 3.21330476887886852338836372284, 3.63375912994371950455679240281, 3.91496065289375490933073148483, 4.83939395504323857616301277266, 4.84147234427561243175964105754, 5.43832227181315514769757423559, 5.71171212591502287184577448974, 6.29418978351181303233847587232, 6.37297320641009731219027408655, 7.27997338241203579758958449557, 7.46650868845705331512986323841, 7.81995041110292028237710070100, 7.84452220382652342846946528697, 8.541485438164292735418107891576, 8.556702936501436208141737692509

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