Properties

Label 4-1050e2-1.1-c1e2-0-38
Degree $4$
Conductor $1102500$
Sign $1$
Analytic cond. $70.2963$
Root an. cond. $2.89556$
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-s − 3-s − 6-s − 7-s − 8-s − 5·11-s − 14-s − 16-s − 4·17-s − 8·19-s + 21-s − 5·22-s − 4·23-s + 24-s + 27-s − 10·29-s − 3·31-s + 5·33-s − 4·34-s − 4·37-s − 8·38-s + 42-s − 4·43-s − 4·46-s − 6·47-s + 48-s − 6·49-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 1.50·11-s − 0.267·14-s − 1/4·16-s − 0.970·17-s − 1.83·19-s + 0.218·21-s − 1.06·22-s − 0.834·23-s + 0.204·24-s + 0.192·27-s − 1.85·29-s − 0.538·31-s + 0.870·33-s − 0.685·34-s − 0.657·37-s − 1.29·38-s + 0.154·42-s − 0.609·43-s − 0.589·46-s − 0.875·47-s + 0.144·48-s − 6/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1102500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(70.2963\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1050} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1102500,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
good11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−9.678689101341295452822122358106, −9.434956655005391004339348037327, −8.810438674498531201628486466349, −8.493348126849116086106028590215, −7.86893710961468314708691687304, −7.82401896633040635985045540793, −6.97977074030033089473429676998, −6.51360396989113378177973354703, −6.41007476745326135587539873599, −5.76410473914077211476771835103, −5.21668572551171480318499492699, −5.13071125691891994709698534688, −4.50595392829906558172204785000, −3.78681309717082734740436612075, −3.73006254594142138554368954655, −2.78962558431298810407728049052, −2.25711565454085477630407995337, −1.78347108183832061810664122247, 0, 0, 1.78347108183832061810664122247, 2.25711565454085477630407995337, 2.78962558431298810407728049052, 3.73006254594142138554368954655, 3.78681309717082734740436612075, 4.50595392829906558172204785000, 5.13071125691891994709698534688, 5.21668572551171480318499492699, 5.76410473914077211476771835103, 6.41007476745326135587539873599, 6.51360396989113378177973354703, 6.97977074030033089473429676998, 7.82401896633040635985045540793, 7.86893710961468314708691687304, 8.493348126849116086106028590215, 8.810438674498531201628486466349, 9.434956655005391004339348037327, 9.678689101341295452822122358106

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