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Properties

Label 4-980e2-1.1-c1e2-0-30
Degree $4$
Conductor $960400$
Sign $1$
Analytic cond. $61.2359$
Root an. cond. $2.79738$
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  − 2·3-s − 5-s + 3·9-s − 4·13-s + 2·15-s − 6·17-s − 4·19-s − 6·23-s − 10·27-s + 12·29-s − 4·31-s − 2·37-s + 8·39-s − 12·41-s − 20·43-s − 3·45-s − 6·47-s + 12·51-s + 6·53-s + 8·57-s + 12·59-s + 2·61-s + 4·65-s − 2·67-s + 12·69-s − 24·71-s + 2·73-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 9-s − 1.10·13-s + 0.516·15-s − 1.45·17-s − 0.917·19-s − 1.25·23-s − 1.92·27-s + 2.22·29-s − 0.718·31-s − 0.328·37-s + 1.28·39-s − 1.87·41-s − 3.04·43-s − 0.447·45-s − 0.875·47-s + 1.68·51-s + 0.824·53-s + 1.05·57-s + 1.56·59-s + 0.256·61-s + 0.496·65-s − 0.244·67-s + 1.44·69-s − 2.84·71-s + 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(960400\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(61.2359\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 960400,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_b
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_t
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_ad
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_n
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.e_ap
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_abh
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.43.u_he
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_al
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_ar
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_dh
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_acf
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_acl
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_acr
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ap
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_acb
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 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

−10.03965015248128628912106807092, −9.609312979669962421036077568725, −8.765618259406640218291553476730, −8.469029290848132533946342079075, −8.346358474554124649413682515239, −7.55063861849093784405400132968, −7.00697775460501886819900456835, −6.96309481843633676754822466877, −6.22388277960908917403967252649, −6.18254081660402112783335453292, −5.27782986439097672900272675997, −5.01832465489147199907528245066, −4.56146916215330830942690266943, −4.12663907617630997213205759291, −3.59689222064108361214940628665, −2.81707654877198533725577674054, −2.05404859426744015397564861568, −1.59507968108844440572611551061, 0, 0, 1.59507968108844440572611551061, 2.05404859426744015397564861568, 2.81707654877198533725577674054, 3.59689222064108361214940628665, 4.12663907617630997213205759291, 4.56146916215330830942690266943, 5.01832465489147199907528245066, 5.27782986439097672900272675997, 6.18254081660402112783335453292, 6.22388277960908917403967252649, 6.96309481843633676754822466877, 7.00697775460501886819900456835, 7.55063861849093784405400132968, 8.346358474554124649413682515239, 8.469029290848132533946342079075, 8.765618259406640218291553476730, 9.609312979669962421036077568725, 10.03965015248128628912106807092

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