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Properties

Label 4-984e2-1.1-c1e2-0-37
Degree $4$
Conductor $968256$
Sign $1$
Analytic cond. $61.7368$
Root an. cond. $2.80308$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4·5-s − 4·7-s + 3·9-s + 2·11-s − 4·13-s − 8·15-s − 2·17-s − 8·19-s − 8·21-s + 4·25-s + 4·27-s − 10·29-s − 10·31-s + 4·33-s + 16·35-s − 10·37-s − 8·39-s + 2·41-s − 14·43-s − 12·45-s − 6·47-s − 4·51-s + 8·53-s − 8·55-s − 16·57-s + 8·59-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s − 1.51·7-s + 9-s + 0.603·11-s − 1.10·13-s − 2.06·15-s − 0.485·17-s − 1.83·19-s − 1.74·21-s + 4/5·25-s + 0.769·27-s − 1.85·29-s − 1.79·31-s + 0.696·33-s + 2.70·35-s − 1.64·37-s − 1.28·39-s + 0.312·41-s − 2.13·43-s − 1.78·45-s − 0.875·47-s − 0.560·51-s + 1.09·53-s − 1.07·55-s − 2.11·57-s + 1.04·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(968256\)    =    \(2^{6} \cdot 3^{2} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(61.7368\)
Root analytic conductor: \(2.80308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 968256,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
41$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_m
7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_q
11$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_f
13$D_{4}$ \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_bc
17$D_{4}$ \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_r
19$D_{4}$ \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.19.i_bk
23$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.23.a_bc
29$D_{4}$ \( 1 + 10 T + 81 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.29.k_dd
31$D_{4}$ \( 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_db
37$D_{4}$ \( 1 + 10 T + 91 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.37.k_dn
43$D_{4}$ \( 1 + 14 T + 127 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.43.o_ex
47$D_{4}$ \( 1 + 6 T + 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_cb
53$D_{4}$ \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_by
59$D_{4}$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_ew
61$D_{4}$ \( 1 + 6 T + 3 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_d
67$D_{4}$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_co
71$D_{4}$ \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.71.c_at
73$D_{4}$ \( 1 - 18 T + 195 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.73.as_hn
79$D_{4}$ \( 1 + 12 T + 66 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_co
83$D_{4}$ \( 1 - 12 T + 200 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_hs
89$D_{4}$ \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_ha
97$D_{4}$ \( 1 - 16 T + 208 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.97.aq_ia
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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

−9.712482892239914191247919075280, −9.282547312402058406108116522287, −8.799973839012882942828041085780, −8.755706551406045936164988684969, −8.026199780558809639640479110323, −7.78207094580209270304159625013, −7.29175752915425266960013868994, −6.90009522037559025041217676680, −6.66980298240299584730354879534, −6.12312120007661530702229236159, −5.31153530000306123334816122702, −4.81559432844023044610460387214, −4.09255488164938693865854402670, −3.88736610053655178589275051522, −3.38183385196436392183072440291, −3.24918813706399216198353444045, −2.09139172604920839441434428079, −1.96560729536139344873715299510, 0, 0, 1.96560729536139344873715299510, 2.09139172604920839441434428079, 3.24918813706399216198353444045, 3.38183385196436392183072440291, 3.88736610053655178589275051522, 4.09255488164938693865854402670, 4.81559432844023044610460387214, 5.31153530000306123334816122702, 6.12312120007661530702229236159, 6.66980298240299584730354879534, 6.90009522037559025041217676680, 7.29175752915425266960013868994, 7.78207094580209270304159625013, 8.026199780558809639640479110323, 8.755706551406045936164988684969, 8.799973839012882942828041085780, 9.282547312402058406108116522287, 9.712482892239914191247919075280

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