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Properties

Label 4-4080e2-1.1-c1e2-0-18
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 − 8·41-s + 4·45-s − 2·49-s − 16·55-s − 12·61-s + 28·71-s + 24·79-s + 81-s + 12·89-s − 32·95-s − 4·99-s + 36·101-s + 28·109-s − 10·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 1.78·5-s − 1/3·9-s + 1.20·11-s + 1.83·19-s + 11/5·25-s − 1.24·41-s + 0.596·45-s − 2/7·49-s − 2.15·55-s − 1.53·61-s + 3.32·71-s + 2.70·79-s + 1/9·81-s + 1.27·89-s − 3.28·95-s − 0.402·99-s + 3.58·101-s + 2.68·109-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 + 0.0819·149-s + 0.0813·151-s + 0.0798·157-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

\(L(1)\) \(\approx\) \(2.230063825\)
\(L(\frac12)\) \(\approx\) \(2.230063825\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.37.a_abm
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.41.i_du
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.67.a_ck
71$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.71.abc_na
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.73.a_afm
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.79.ay_lq
83$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.83.a_agk
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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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

−8.456655380178006661305438056661, −8.312652414596324046369567633254, −7.69963161463794043404834309220, −7.69139832716121545605024819687, −7.18223850569773842799299829895, −6.93899423123143280815450997270, −6.38031687293512113141813939546, −6.22888030897461407755667160101, −5.64793081877995870388921830321, −4.99481368616522063719303160031, −4.87336352452533502553195851085, −4.58370893399363191322991856015, −3.73851389650504173593690128698, −3.65384864577707607386444782508, −3.39571806270551857018146246758, −2.97113872924937310279157185769, −2.19819011078181399641697313788, −1.66005613079495279951556790065, −0.811320748992372361741001669583, −0.62355283679895436373638912105, 0.62355283679895436373638912105, 0.811320748992372361741001669583, 1.66005613079495279951556790065, 2.19819011078181399641697313788, 2.97113872924937310279157185769, 3.39571806270551857018146246758, 3.65384864577707607386444782508, 3.73851389650504173593690128698, 4.58370893399363191322991856015, 4.87336352452533502553195851085, 4.99481368616522063719303160031, 5.64793081877995870388921830321, 6.22888030897461407755667160101, 6.38031687293512113141813939546, 6.93899423123143280815450997270, 7.18223850569773842799299829895, 7.69139832716121545605024819687, 7.69963161463794043404834309220, 8.312652414596324046369567633254, 8.456655380178006661305438056661

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