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Properties

Label 4-6192e2-1.1-c1e2-0-6
Degree $4$
Conductor $38340864$
Sign $1$
Analytic cond. $2444.64$
Root an. cond. $7.03159$
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 + 2·25-s − 12·29-s + 4·31-s + 10·43-s + 14·49-s − 28·67-s + 24·71-s − 20·79-s + 28·89-s − 4·97-s + 12·109-s − 20·113-s + 20·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s − 48·145-s + 149-s + 151-s + 16·155-s + 157-s + 163-s + 167-s − 26·169-s + ⋯
L(s)  = 1  + 1.78·5-s + 2/5·25-s − 2.22·29-s + 0.718·31-s + 1.52·43-s + 2·49-s − 3.42·67-s + 2.84·71-s − 2.25·79-s + 2.96·89-s − 0.406·97-s + 1.14·109-s − 1.88·113-s + 1.81·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.98·145-s + 0.0819·149-s + 0.0813·151-s + 1.28·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38340864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38340864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(38340864\)    =    \(2^{8} \cdot 3^{4} \cdot 43^{2}\)
Sign: $1$
Analytic conductor: \(2444.64\)
Root analytic conductor: \(7.03159\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 38340864,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.536205791\)
\(L(\frac12)\) \(\approx\) \(3.536205791\)
\(L(\frac{3}{2})\) 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 \( 1 \)
43$C_2$ \( 1 - 10 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.5.ae_o
7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.11.a_au
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.17.a_q
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.19.a_ag
23$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.23.a_abs
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.37.a_ac
41$C_2^2$ \( 1 - 64 T^{2} + p^{2} T^{4} \) 2.41.a_acm
47$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \) 2.47.a_ado
53$C_2^2$ \( 1 - 104 T^{2} + p^{2} T^{4} \) 2.53.a_aea
59$C_2^2$ \( 1 - 100 T^{2} + p^{2} T^{4} \) 2.59.a_adw
61$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.61.a_aek
67$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.67.bc_ms
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.73.a_acw
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.83.a_ae
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.89.abc_ok
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−8.242405662030258824856328644271, −7.61749129915978039315507029499, −7.52689710020325060454207058230, −7.39426881815046448226402060003, −6.64744606947532338204536191380, −6.32500859603736805387911443889, −6.15018555018091420453494127223, −5.67405971888223648075388446161, −5.45292393126819028198185801086, −5.29136155986419149155434521765, −4.61581222742678999292596709060, −4.19385046811841635990204175365, −3.90782513326486043158528606965, −3.41498801087719712434876087439, −2.82392068479914678970395537547, −2.45208836963622070017432694429, −2.06019643214029986820398866131, −1.71704031727468607771081389797, −1.19397012421479630073385174867, −0.46656058733173418772523303150, 0.46656058733173418772523303150, 1.19397012421479630073385174867, 1.71704031727468607771081389797, 2.06019643214029986820398866131, 2.45208836963622070017432694429, 2.82392068479914678970395537547, 3.41498801087719712434876087439, 3.90782513326486043158528606965, 4.19385046811841635990204175365, 4.61581222742678999292596709060, 5.29136155986419149155434521765, 5.45292393126819028198185801086, 5.67405971888223648075388446161, 6.15018555018091420453494127223, 6.32500859603736805387911443889, 6.64744606947532338204536191380, 7.39426881815046448226402060003, 7.52689710020325060454207058230, 7.61749129915978039315507029499, 8.242405662030258824856328644271

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