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Properties

Label 4-4080e2-1.1-c1e2-0-23
Degree $4$
Conductor $16646400$
Sign $1$
Analytic cond. $1061.38$
Root an. cond. $5.70779$
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  + 4·5-s − 9-s − 4·11-s − 8·19-s + 11·25-s + 16·29-s + 8·31-s − 4·45-s + 14·49-s − 16·55-s + 8·59-s + 20·61-s + 12·71-s + 32·79-s + 81-s + 12·89-s − 32·95-s + 4·99-s − 4·101-s + 12·109-s − 10·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + ⋯
L(s)  = 1  + 1.78·5-s − 1/3·9-s − 1.20·11-s − 1.83·19-s + 11/5·25-s + 2.97·29-s + 1.43·31-s − 0.596·45-s + 2·49-s − 2.15·55-s + 1.04·59-s + 2.56·61-s + 1.42·71-s + 3.60·79-s + 1/9·81-s + 1.27·89-s − 3.28·95-s + 0.402·99-s − 0.398·101-s + 1.14·109-s − 0.909·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16646400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16646400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16646400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1061.38\)
Root analytic conductor: \(5.70779\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16646400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.542866921\)
\(L(\frac12)\) \(\approx\) \(4.542866921\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
17$C_2$ \( 1 + T^{2} \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.29.aq_es
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.43.a_ade
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.53.a_ady
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.67.a_abi
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.79.abg_py
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.97.a_adq
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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

−8.538043590169387456704371704897, −8.357529104628115459573030187177, −8.018637310478987892033370438127, −7.63617593435672316809695127655, −6.85868350287853257196638274739, −6.64296194617510584936766938181, −6.45945106189269409065848371080, −6.16795502775914991868858555203, −5.47973309338948041551610675089, −5.44089513371555716876276712889, −4.83298792426919868329053953347, −4.73576131756657731109615316811, −4.09725575947784650591375236852, −3.61090060798732348258679260187, −2.87067060393959220606170846166, −2.59141248744084903569790550375, −2.19108560478946047283494081559, −2.08628701764427572998960436329, −0.880479173623765391177071728144, −0.78860445711744050515473495769, 0.78860445711744050515473495769, 0.880479173623765391177071728144, 2.08628701764427572998960436329, 2.19108560478946047283494081559, 2.59141248744084903569790550375, 2.87067060393959220606170846166, 3.61090060798732348258679260187, 4.09725575947784650591375236852, 4.73576131756657731109615316811, 4.83298792426919868329053953347, 5.44089513371555716876276712889, 5.47973309338948041551610675089, 6.16795502775914991868858555203, 6.45945106189269409065848371080, 6.64296194617510584936766938181, 6.85868350287853257196638274739, 7.63617593435672316809695127655, 8.018637310478987892033370438127, 8.357529104628115459573030187177, 8.538043590169387456704371704897

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