Properties

Label 4-2850e2-1.1-c1e2-0-14
Degree $4$
Conductor $8122500$
Sign $1$
Analytic cond. $517.897$
Root an. cond. $4.77046$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s − 8·11-s + 16-s + 2·19-s + 4·29-s + 8·31-s + 36-s + 20·41-s + 8·44-s + 14·49-s − 24·59-s + 28·61-s − 64-s + 16·71-s − 2·76-s + 8·79-s + 81-s + 12·89-s + 8·99-s + 4·101-s + 12·109-s − 4·116-s + 26·121-s − 8·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 2.41·11-s + 1/4·16-s + 0.458·19-s + 0.742·29-s + 1.43·31-s + 1/6·36-s + 3.12·41-s + 1.20·44-s + 2·49-s − 3.12·59-s + 3.58·61-s − 1/8·64-s + 1.89·71-s − 0.229·76-s + 0.900·79-s + 1/9·81-s + 1.27·89-s + 0.804·99-s + 0.398·101-s + 1.14·109-s − 0.371·116-s + 2.36·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8122500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(517.897\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.028373536\)
\(L(\frac12)\) \(\approx\) \(2.028373536\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−8.920988396000801812360307286564, −8.552401287104038513288961730484, −8.055573476983164953450807104501, −7.965566884637014119472543713347, −7.59214014436594803960150618313, −7.26062068944332663136375686407, −6.65335616001666438648455720278, −6.27716716059072206147969104682, −5.67278318263461060108272592286, −5.60158468053906102238467638373, −5.13618679792012335541814618000, −4.61499983890732478160884711074, −4.47345439038734654798910190599, −3.79894750409145263244343700128, −3.29343566030573104969788660387, −2.66436393575797093320308151119, −2.57104854456918517039530813935, −2.03633990584666325228816963803, −0.76155585672793965265685803370, −0.69195047505261628641561516251, 0.69195047505261628641561516251, 0.76155585672793965265685803370, 2.03633990584666325228816963803, 2.57104854456918517039530813935, 2.66436393575797093320308151119, 3.29343566030573104969788660387, 3.79894750409145263244343700128, 4.47345439038734654798910190599, 4.61499983890732478160884711074, 5.13618679792012335541814618000, 5.60158468053906102238467638373, 5.67278318263461060108272592286, 6.27716716059072206147969104682, 6.65335616001666438648455720278, 7.26062068944332663136375686407, 7.59214014436594803960150618313, 7.965566884637014119472543713347, 8.055573476983164953450807104501, 8.552401287104038513288961730484, 8.920988396000801812360307286564

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