Properties

Label 4-2736e2-1.1-c2e2-0-6
Degree $4$
Conductor $7485696$
Sign $1$
Analytic cond. $5557.79$
Root an. cond. $8.63426$
Motivic weight $2$
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  + 8·5-s + 2·7-s + 28·11-s − 46·17-s − 20·19-s − 2·23-s − 2·25-s + 16·35-s − 136·43-s + 52·47-s − 95·49-s + 224·55-s − 80·61-s − 14·73-s + 56·77-s + 64·83-s − 368·85-s − 160·95-s − 28·101-s − 16·115-s − 92·119-s + 346·121-s − 344·125-s + 127-s + 131-s − 40·133-s + 137-s + ⋯
L(s)  = 1  + 8/5·5-s + 2/7·7-s + 2.54·11-s − 2.70·17-s − 1.05·19-s − 0.0869·23-s − 0.0799·25-s + 0.457·35-s − 3.16·43-s + 1.10·47-s − 1.93·49-s + 4.07·55-s − 1.31·61-s − 0.191·73-s + 8/11·77-s + 0.771·83-s − 4.32·85-s − 1.68·95-s − 0.277·101-s − 0.139·115-s − 0.773·119-s + 2.85·121-s − 2.75·125-s + 0.00787·127-s + 0.00763·131-s − 0.300·133-s + 0.00729·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7485696\)    =    \(2^{8} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5557.79\)
Root analytic conductor: \(8.63426\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2736} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7485696,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.282418706\)
\(L(\frac12)\) \(\approx\) \(2.282418706\)
\(L(2)\) 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)$
bad2 \( 1 \)
3 \( 1 \)
19$C_2$ \( 1 + 20 T + p^{2} T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p^{2} T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 77 T^{2} + p^{4} T^{4} \)
17$C_2$ \( ( 1 + 23 T + p^{2} T^{2} )^{2} \)
23$C_2$ \( ( 1 + T + p^{2} T^{2} )^{2} \)
29$C_2^2$ \( 1 + 23 p T^{2} + p^{4} T^{4} \)
31$C_2^2$ \( 1 - 878 T^{2} + p^{4} T^{4} \)
37$C_2^2$ \( 1 - 1694 T^{2} + p^{4} T^{4} \)
41$C_2^2$ \( 1 - 2318 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 68 T + p^{2} T^{2} )^{2} \)
47$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )^{2} \)
53$C_2^2$ \( 1 + 907 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 6701 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 40 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 8717 T^{2} + p^{4} T^{4} \)
71$C_2^2$ \( 1 - 9038 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 7 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 3086 T^{2} + p^{4} T^{4} \)
83$C_2$ \( ( 1 - 32 T + p^{2} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 862 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 9422 T^{2} + p^{4} T^{4} \)
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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

−9.092292073977344619226420537078, −8.431785666303300060853522271220, −8.340608077545456062143917821829, −7.80263981184762920350889226268, −6.93712963739735176331479308540, −6.79921111500720999972393408852, −6.58560564364407808558447519741, −6.20184420731952870582747070259, −5.99832067404944156345883922116, −5.41738454088237330279949689512, −4.85569411652833833339777076578, −4.36591277665851507663040142204, −4.33051914334330993187910979470, −3.65400151638158908060884796036, −3.23408935616517252793460482226, −2.39601033317843004044736227317, −2.01974818398281076506408784461, −1.66059932331320012918791253669, −1.43261392104023903264379396732, −0.31635098721583418729892834707, 0.31635098721583418729892834707, 1.43261392104023903264379396732, 1.66059932331320012918791253669, 2.01974818398281076506408784461, 2.39601033317843004044736227317, 3.23408935616517252793460482226, 3.65400151638158908060884796036, 4.33051914334330993187910979470, 4.36591277665851507663040142204, 4.85569411652833833339777076578, 5.41738454088237330279949689512, 5.99832067404944156345883922116, 6.20184420731952870582747070259, 6.58560564364407808558447519741, 6.79921111500720999972393408852, 6.93712963739735176331479308540, 7.80263981184762920350889226268, 8.340608077545456062143917821829, 8.431785666303300060853522271220, 9.092292073977344619226420537078

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