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Properties

Label 4-80e4-1.1-c1e2-0-10
Degree $4$
Conductor $40960000$
Sign $1$
Analytic cond. $2611.64$
Root an. cond. $7.14872$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 3·9-s − 6·11-s − 6·17-s + 2·19-s − 14·27-s − 12·33-s + 18·41-s + 8·43-s − 2·49-s − 12·51-s + 4·57-s + 24·59-s + 22·67-s + 14·73-s − 4·81-s + 30·83-s + 6·89-s − 28·97-s + 18·99-s + 18·107-s + 30·113-s + 5·121-s + 36·123-s + 127-s + 16·129-s + 131-s + ⋯
L(s)  = 1  + 1.15·3-s − 9-s − 1.80·11-s − 1.45·17-s + 0.458·19-s − 2.69·27-s − 2.08·33-s + 2.81·41-s + 1.21·43-s − 2/7·49-s − 1.68·51-s + 0.529·57-s + 3.12·59-s + 2.68·67-s + 1.63·73-s − 4/9·81-s + 3.29·83-s + 0.635·89-s − 2.84·97-s + 1.80·99-s + 1.74·107-s + 2.82·113-s + 5/11·121-s + 3.24·123-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40960000\)    =    \(2^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2611.64\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 40960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.120830940\)
\(L(\frac12)\) \(\approx\) \(3.120830940\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.3.ac_h
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.19.ac_bn
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.29.a_aby
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.31.a_o
37$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.37.a_abi
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.41.as_gh
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.47.a_ao
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.61.a_eg
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.67.aw_jv
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.71.a_bi
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.73.ao_hn
79$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.79.a_by
83$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.83.abe_pb
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.89.ag_hf
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_pa
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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

−8.121551486404321807579172298252, −7.971506485743524099606670889883, −7.66669711066692566948427417035, −7.33029313194387116859238151967, −6.70826783155354939155402973149, −6.65659579724916665182220986785, −5.81669926677120065769032818369, −5.78939825592860223729973557116, −5.43833471174017576787374047083, −4.98864941695025249747637790290, −4.62375148092207638761282932358, −4.12792276318147290954703909741, −3.53613629705261095516449092719, −3.52982967630848746479223576614, −2.79544603177310749832203885790, −2.43788882325287122092596083281, −2.24775093338849679260341772685, −2.11992516089796405376085111989, −0.74679275324074692525083658792, −0.54993784084292537487854106048, 0.54993784084292537487854106048, 0.74679275324074692525083658792, 2.11992516089796405376085111989, 2.24775093338849679260341772685, 2.43788882325287122092596083281, 2.79544603177310749832203885790, 3.52982967630848746479223576614, 3.53613629705261095516449092719, 4.12792276318147290954703909741, 4.62375148092207638761282932358, 4.98864941695025249747637790290, 5.43833471174017576787374047083, 5.78939825592860223729973557116, 5.81669926677120065769032818369, 6.65659579724916665182220986785, 6.70826783155354939155402973149, 7.33029313194387116859238151967, 7.66669711066692566948427417035, 7.971506485743524099606670889883, 8.121551486404321807579172298252

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