Properties

Label 4-2664e2-1.1-c1e2-0-3
Degree $4$
Conductor $7096896$
Sign $1$
Analytic cond. $452.504$
Root an. cond. $4.61217$
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 − 4·7-s − 4·13-s + 4·17-s + 8·23-s + 4·25-s + 12·29-s − 8·31-s − 16·35-s − 2·37-s + 12·41-s + 6·49-s + 20·53-s + 4·61-s − 16·65-s + 4·67-s − 16·71-s + 16·73-s + 16·79-s − 16·83-s + 16·85-s + 12·89-s + 16·91-s + 4·97-s − 4·101-s − 8·103-s − 16·107-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.51·7-s − 1.10·13-s + 0.970·17-s + 1.66·23-s + 4/5·25-s + 2.22·29-s − 1.43·31-s − 2.70·35-s − 0.328·37-s + 1.87·41-s + 6/7·49-s + 2.74·53-s + 0.512·61-s − 1.98·65-s + 0.488·67-s − 1.89·71-s + 1.87·73-s + 1.80·79-s − 1.75·83-s + 1.73·85-s + 1.27·89-s + 1.67·91-s + 0.406·97-s − 0.398·101-s − 0.788·103-s − 1.54·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7096896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7096896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7096896\)    =    \(2^{6} \cdot 3^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(452.504\)
Root analytic conductor: \(4.61217\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7096896,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.173374301\)
\(L(\frac12)\) \(\approx\) \(3.173374301\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.5.ae_m
7$C_4$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_k
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_w
17$D_{4}$ \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_bk
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.19.a_g
23$D_{4}$ \( 1 - 8 T + 60 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_ci
29$D_{4}$ \( 1 - 12 T + 76 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.29.am_cy
31$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_cs
41$C_4$ \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.41.am_di
43$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.43.a_da
47$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.47.a_ck
53$D_{4}$ \( 1 - 20 T + 198 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.53.au_hq
59$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \) 2.59.a_dw
61$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_ac
67$D_{4}$ \( 1 - 4 T + 130 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_fa
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.73.aq_ic
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.79.aq_hi
83$D_{4}$ \( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.83.q_gc
89$D_{4}$ \( 1 - 12 T + 164 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_gi
97$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_ac
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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

−9.173600693372147938682937301986, −8.906871857739975549094719065495, −8.335147139073202048586367610641, −7.922810962306191120465967733074, −7.25758607612397101262315506497, −7.14874486987630258692906760413, −6.69793747069327430172809301451, −6.37860711243168680742285091339, −5.95402724350124468029955057013, −5.53413353430735119121269351325, −5.30817654674783078546262725132, −4.96904902888304444661653108831, −4.21220662622938707632058833731, −3.86713175420322982562519143319, −3.05810897881563679597266210612, −2.95372989549018036523062830306, −2.39776509711746329883742241275, −2.03065987467896487125104360531, −1.15158556770127374021296500838, −0.63611373220484911698743267493, 0.63611373220484911698743267493, 1.15158556770127374021296500838, 2.03065987467896487125104360531, 2.39776509711746329883742241275, 2.95372989549018036523062830306, 3.05810897881563679597266210612, 3.86713175420322982562519143319, 4.21220662622938707632058833731, 4.96904902888304444661653108831, 5.30817654674783078546262725132, 5.53413353430735119121269351325, 5.95402724350124468029955057013, 6.37860711243168680742285091339, 6.69793747069327430172809301451, 7.14874486987630258692906760413, 7.25758607612397101262315506497, 7.922810962306191120465967733074, 8.335147139073202048586367610641, 8.906871857739975549094719065495, 9.173600693372147938682937301986

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