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Properties

Label 4-1134e2-1.1-c1e2-0-30
Degree $4$
Conductor $1285956$
Sign $1$
Analytic cond. $81.9936$
Root an. cond. $3.00915$
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  + 2-s − 6·5-s − 4·7-s − 8-s − 6·10-s + 6·11-s − 2·13-s − 4·14-s − 16-s + 6·17-s − 2·19-s + 6·22-s + 12·23-s + 17·25-s − 2·26-s + 9·29-s + 7·31-s + 6·34-s + 24·35-s + 10·37-s − 2·38-s + 6·40-s + 4·43-s + 12·46-s + 12·47-s + 9·49-s + 17·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 2.68·5-s − 1.51·7-s − 0.353·8-s − 1.89·10-s + 1.80·11-s − 0.554·13-s − 1.06·14-s − 1/4·16-s + 1.45·17-s − 0.458·19-s + 1.27·22-s + 2.50·23-s + 17/5·25-s − 0.392·26-s + 1.67·29-s + 1.25·31-s + 1.02·34-s + 4.05·35-s + 1.64·37-s − 0.324·38-s + 0.948·40-s + 0.609·43-s + 1.76·46-s + 1.75·47-s + 9/7·49-s + 2.40·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1285956\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(81.9936\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.690662146\)
\(L(\frac12)\) \(\approx\) \(1.690662146\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.11.ag_bf
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.c_aj
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_t
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ap
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_ca
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.ah_s
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.37.ak_cl
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_abb
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_dt
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_abs
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.59.d_aby
61$C_2^2$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_abt
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_acl
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_acr
79$C_2^2$ \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.79.f_acc
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.83.aj_ac
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_acb
97$C_2^2$ \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.97.an_cu
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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

−9.787139695390997440403493160242, −9.726054653558983734764329727628, −9.070680312802161235305342108040, −8.851647777639541928662659320889, −8.353231050658178692623954606532, −7.85397161044719272567085924444, −7.52661575484205842675603170925, −7.06952485579367031972517453728, −6.63261219628726278398161055772, −6.48825635455924513510815005127, −5.81213056561974991646725969432, −5.21324057022628905359156653046, −4.49991948914571717453111671184, −4.24487051295829938467574555157, −4.02488429265996079437928500239, −3.38669133562200595296484428438, −2.96768067327223288966911017286, −2.82458963834384751437891636840, −0.975484638579017762636638776897, −0.70598724177145393018985165130, 0.70598724177145393018985165130, 0.975484638579017762636638776897, 2.82458963834384751437891636840, 2.96768067327223288966911017286, 3.38669133562200595296484428438, 4.02488429265996079437928500239, 4.24487051295829938467574555157, 4.49991948914571717453111671184, 5.21324057022628905359156653046, 5.81213056561974991646725969432, 6.48825635455924513510815005127, 6.63261219628726278398161055772, 7.06952485579367031972517453728, 7.52661575484205842675603170925, 7.85397161044719272567085924444, 8.353231050658178692623954606532, 8.851647777639541928662659320889, 9.070680312802161235305342108040, 9.726054653558983734764329727628, 9.787139695390997440403493160242

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