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Properties

Label 4-1008e2-1.1-c1e2-0-144
Degree $4$
Conductor $1016064$
Sign $1$
Analytic cond. $64.7851$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s − 3·9-s − 3·11-s − 13-s − 3·17-s + 5·19-s + 23-s + 5·25-s − 9·29-s − 8·31-s + 4·35-s − 5·37-s − 7·41-s + 3·43-s + 3·45-s − 16·47-s + 9·49-s − 9·53-s + 3·55-s + 8·59-s + 4·61-s + 12·63-s + 65-s − 24·67-s − 16·71-s + 13·73-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s − 9-s − 0.904·11-s − 0.277·13-s − 0.727·17-s + 1.14·19-s + 0.208·23-s + 25-s − 1.67·29-s − 1.43·31-s + 0.676·35-s − 0.821·37-s − 1.09·41-s + 0.457·43-s + 0.447·45-s − 2.33·47-s + 9/7·49-s − 1.23·53-s + 0.404·55-s + 1.04·59-s + 0.512·61-s + 1.51·63-s + 0.124·65-s − 2.93·67-s − 1.89·71-s + 1.52·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(64.7851\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1016064,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ae
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) 2.13.b_am
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.19.af_g
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) 2.23.ab_aw
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.29.j_ca
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2^2$ \( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.37.f_am
41$C_2^2$ \( 1 + 7 T + 8 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.41.h_i
43$C_2^2$ \( 1 - 3 T - 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.43.ad_abi
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.53.j_bc
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2^2$ \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.73.an_ds
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2^2$ \( 1 + 13 T + 86 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.83.n_di
89$C_2^2$ \( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.89.aj_ai
97$C_2^2$ \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.97.ar_hk
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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

−9.640813084017751884046979873691, −9.293786222245745217116822390839, −9.082907810347376959998651048511, −8.489639522096411240271802810640, −8.225002472532366261414108130122, −7.50533697208485644202273568578, −7.13374651509719037938873481749, −7.07852226127769372973605577338, −6.12914338337346937117504245595, −6.08625612207211162835708968721, −5.35878937451834078505323644327, −5.08296974785184007746066599780, −4.54339003012016005954261696100, −3.65707512955832332916819981104, −3.30769056500662343019213428366, −3.06459731335624296264540224880, −2.38635645349822569884973618318, −1.56125568281334883019518887480, 0, 0, 1.56125568281334883019518887480, 2.38635645349822569884973618318, 3.06459731335624296264540224880, 3.30769056500662343019213428366, 3.65707512955832332916819981104, 4.54339003012016005954261696100, 5.08296974785184007746066599780, 5.35878937451834078505323644327, 6.08625612207211162835708968721, 6.12914338337346937117504245595, 7.07852226127769372973605577338, 7.13374651509719037938873481749, 7.50533697208485644202273568578, 8.225002472532366261414108130122, 8.489639522096411240271802810640, 9.082907810347376959998651048511, 9.293786222245745217116822390839, 9.640813084017751884046979873691

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