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Properties

Label 4-756e2-1.1-c1e2-0-10
Degree $4$
Conductor $571536$
Sign $1$
Analytic cond. $36.4416$
Root an. cond. $2.45696$
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  + 5·7-s − 10·25-s − 2·37-s + 16·43-s + 18·49-s + 22·67-s − 26·79-s + 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 23·169-s + 173-s − 50·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 1.88·7-s − 2·25-s − 0.328·37-s + 2.43·43-s + 18/7·49-s + 2.68·67-s − 2.92·79-s + 0.383·109-s − 2·121-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 + 0.0783·163-s + 0.0773·167-s + 1.76·169-s + 0.0760·173-s − 3.77·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(571536\)    =    \(2^{4} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(36.4416\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 571536,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.432512266\)
\(L(\frac12)\) \(\approx\) \(2.432512266\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.a_ax
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.a_bl
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.37.c_cx
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.a_abv
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.67.aw_jv
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.73.a_afn
79$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.79.ba_mp
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.97.a_gn
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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

−8.321813700481147925892685842672, −8.079851394884847745055740317792, −7.51552880521900195309523002532, −7.31346310196346190295346056081, −6.70078665608916533782840710374, −5.98139804295706451305092616431, −5.49030247404272458638604331312, −5.41619595580815395232018816358, −4.44194996130619826314058940000, −4.33573970367631074650890566231, −3.77813458199186956523245005153, −2.91173766719274595341337420733, −2.12496210928289498867796346468, −1.79680621722010121991203363355, −0.833659174677608845252044473607, 0.833659174677608845252044473607, 1.79680621722010121991203363355, 2.12496210928289498867796346468, 2.91173766719274595341337420733, 3.77813458199186956523245005153, 4.33573970367631074650890566231, 4.44194996130619826314058940000, 5.41619595580815395232018816358, 5.49030247404272458638604331312, 5.98139804295706451305092616431, 6.70078665608916533782840710374, 7.31346310196346190295346056081, 7.51552880521900195309523002532, 8.079851394884847745055740317792, 8.321813700481147925892685842672

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