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Properties

Label 4-3328e2-1.1-c1e2-0-7
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  − 2·3-s − 2·5-s + 6·7-s − 9-s − 4·11-s − 2·13-s + 4·15-s + 2·17-s − 12·21-s + 12·23-s + 25-s + 6·27-s + 4·29-s − 4·31-s + 8·33-s − 12·35-s − 14·37-s + 4·39-s + 8·41-s + 14·43-s + 2·45-s + 10·47-s + 15·49-s − 4·51-s + 4·53-s + 8·55-s − 16·59-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 2.26·7-s − 1/3·9-s − 1.20·11-s − 0.554·13-s + 1.03·15-s + 0.485·17-s − 2.61·21-s + 2.50·23-s + 1/5·25-s + 1.15·27-s + 0.742·29-s − 0.718·31-s + 1.39·33-s − 2.02·35-s − 2.30·37-s + 0.640·39-s + 1.24·41-s + 2.13·43-s + 0.298·45-s + 1.45·47-s + 15/7·49-s − 0.560·51-s + 0.549·53-s + 1.07·55-s − 2.08·59-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

\(L(1)\) \(\approx\) \(1.366100496\)
\(L(\frac12)\) \(\approx\) \(1.366100496\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_f
5$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_d
7$D_{4}$ \( 1 - 6 T + 3 p T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.7.ag_v
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_s
17$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.17.ac_bj
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.19.a_g
23$C_4$ \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.23.am_cw
29$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_be
31$D_{4}$ \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_bi
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.37.o_et
41$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.41.ai_co
43$D_{4}$ \( 1 - 14 T + 133 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.43.ao_fd
47$D_{4}$ \( 1 - 10 T + 101 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.47.ak_dx
53$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_da
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.59.q_ha
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$D_{4}$ \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_gg
71$D_{4}$ \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.71.ac_at
73$D_{4}$ \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_fa
79$D_{4}$ \( 1 + 4 T - 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.79.e_abm
83$D_{4}$ \( 1 - 4 T - 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_abe
89$D_{4}$ \( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.89.aq_ic
97$D_{4}$ \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_gk
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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

−8.758064539530912324148640529626, −8.610821912189475961278961746741, −7.76274096674665834953298325581, −7.56744055664395896508286018276, −7.56629676593892581009407910043, −7.24203978358137440590505313893, −6.51070934925789316025165044677, −6.06916666069185765720303208900, −5.58234990639318565746636160697, −5.33776552483667442058432766992, −4.98007214874741965981735348111, −4.81457128932651728844275146505, −4.40457940797217116612463190000, −3.89886203516659175225079807684, −3.08122411075702329249883067410, −2.92293303754370579272386865359, −2.26372996539020685859188889576, −1.66029879489588885104772795668, −0.963694288477816318568717285644, −0.47999263220662198682048470789, 0.47999263220662198682048470789, 0.963694288477816318568717285644, 1.66029879489588885104772795668, 2.26372996539020685859188889576, 2.92293303754370579272386865359, 3.08122411075702329249883067410, 3.89886203516659175225079807684, 4.40457940797217116612463190000, 4.81457128932651728844275146505, 4.98007214874741965981735348111, 5.33776552483667442058432766992, 5.58234990639318565746636160697, 6.06916666069185765720303208900, 6.51070934925789316025165044677, 7.24203978358137440590505313893, 7.56629676593892581009407910043, 7.56744055664395896508286018276, 7.76274096674665834953298325581, 8.610821912189475961278961746741, 8.758064539530912324148640529626

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