Properties

Label 4-1900e2-1.1-c2e2-0-0
Degree $4$
Conductor $3610000$
Sign $1$
Analytic cond. $2680.26$
Root an. cond. $7.19522$
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  + 2·7-s − 11·9-s + 28·11-s − 46·17-s + 20·19-s + 2·23-s − 136·43-s − 52·47-s − 95·49-s − 80·61-s − 22·63-s + 14·73-s + 56·77-s + 40·81-s − 64·83-s − 308·99-s + 28·101-s − 92·119-s + 346·121-s + 127-s + 131-s + 40·133-s + 137-s + 139-s + 149-s + 151-s + 506·153-s + ⋯
L(s)  = 1  + 2/7·7-s − 1.22·9-s + 2.54·11-s − 2.70·17-s + 1.05·19-s + 2/23·23-s − 3.16·43-s − 1.10·47-s − 1.93·49-s − 1.31·61-s − 0.349·63-s + 0.191·73-s + 8/11·77-s + 0.493·81-s − 0.771·83-s − 3.11·99-s + 0.277·101-s − 0.773·119-s + 2.85·121-s + 0.00787·127-s + 0.00763·131-s + 0.300·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 3.30·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3610000\)    =    \(2^{4} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(2680.26\)
Root analytic conductor: \(7.19522\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1900} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3610000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8656664588\)
\(L(\frac12)\) \(\approx\) \(0.8656664588\)
\(L(2)\) 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 \( 1 \)
5 \( 1 \)
19$C_2$ \( 1 - 20 T + p^{2} T^{2} \)
good3$C_2^2$ \( 1 + 11 T^{2} + p^{4} T^{4} \)
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

−9.437091758882549098576925397707, −8.940141667813688088281223377477, −8.522502019931019057223303193042, −8.185332624376198663458020955672, −7.80008402404225782701676376365, −6.91654640209174289938154260310, −6.87951244473126057049337685813, −6.49143340267224120843207149295, −6.22422608353853491382922903383, −5.70853333100271228607214661923, −5.03816048119746040433090629646, −4.75698720140338544383408796040, −4.35460927806022191141160858860, −3.83464977360317413857383659072, −3.16814915360480175547527920534, −3.14958904491427795788910584467, −2.14686116736693669525550679826, −1.70826750013309485902982299127, −1.28278985330950208206832815018, −0.23825991601652280007690044659, 0.23825991601652280007690044659, 1.28278985330950208206832815018, 1.70826750013309485902982299127, 2.14686116736693669525550679826, 3.14958904491427795788910584467, 3.16814915360480175547527920534, 3.83464977360317413857383659072, 4.35460927806022191141160858860, 4.75698720140338544383408796040, 5.03816048119746040433090629646, 5.70853333100271228607214661923, 6.22422608353853491382922903383, 6.49143340267224120843207149295, 6.87951244473126057049337685813, 6.91654640209174289938154260310, 7.80008402404225782701676376365, 8.185332624376198663458020955672, 8.522502019931019057223303193042, 8.940141667813688088281223377477, 9.437091758882549098576925397707

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