Properties

Label 4-2850e2-1.1-c1e2-0-17
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s + 8·11-s + 16-s + 2·19-s − 12·29-s + 8·31-s + 36-s + 20·41-s − 8·44-s + 14·49-s + 24·59-s − 4·61-s − 64-s + 16·71-s − 2·76-s + 8·79-s + 81-s − 20·89-s − 8·99-s + 20·101-s + 12·109-s + 12·116-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 − 2.22·29-s + 1.43·31-s + 1/6·36-s + 3.12·41-s − 1.20·44-s + 2·49-s + 3.12·59-s − 0.512·61-s − 1/8·64-s + 1.89·71-s − 0.229·76-s + 0.900·79-s + 1/9·81-s − 2.11·89-s − 0.804·99-s + 1.99·101-s + 1.14·109-s + 1.11·116-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

\(L(1)\) \(\approx\) \(3.276603408\)
\(L(\frac12)\) \(\approx\) \(3.276603408\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
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$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−8.937411227781701973129571562601, −8.833965476724837117413008451687, −8.267413896694576318783735522270, −7.86037906367919305850001206604, −7.35752364706309258162195544685, −7.21758120764841977917678759360, −6.60228428996643886699393067599, −6.36182086690218974639245422670, −5.84528558062778653975451560080, −5.61592551294927645941695704551, −5.17264045444295940907439437345, −4.54228949037610847675954638480, −4.10370906250141825011030795705, −3.79421947765328079982345185482, −3.70210682105875246603949897418, −2.83315462286494875820042924471, −2.35682492344967270602662253752, −1.80072908839523862312797285160, −0.924079573645545502987278360893, −0.804953668745430946328674224290, 0.804953668745430946328674224290, 0.924079573645545502987278360893, 1.80072908839523862312797285160, 2.35682492344967270602662253752, 2.83315462286494875820042924471, 3.70210682105875246603949897418, 3.79421947765328079982345185482, 4.10370906250141825011030795705, 4.54228949037610847675954638480, 5.17264045444295940907439437345, 5.61592551294927645941695704551, 5.84528558062778653975451560080, 6.36182086690218974639245422670, 6.60228428996643886699393067599, 7.21758120764841977917678759360, 7.35752364706309258162195544685, 7.86037906367919305850001206604, 8.267413896694576318783735522270, 8.833965476724837117413008451687, 8.937411227781701973129571562601

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