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Properties

Label 4-825e2-1.1-c0e2-0-0
Degree $4$
Conductor $680625$
Sign $1$
Analytic cond. $0.169520$
Root an. cond. $0.641660$
Motivic weight $0$
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  − 9-s − 16-s + 2·23-s − 2·37-s + 2·47-s − 2·53-s + 4·59-s + 2·67-s + 81-s + 2·97-s + 2·103-s + 2·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 9-s − 16-s + 2·23-s − 2·37-s + 2·47-s − 2·53-s + 4·59-s + 2·67-s + 81-s + 2·97-s + 2·103-s + 2·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(680625\)    =    \(3^{2} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.169520\)
Root analytic conductor: \(0.641660\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 680625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8143802648\)
\(L(\frac12)\) \(\approx\) \(0.8143802648\)
\(L(1)\) 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)$
bad3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
11$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
59$C_1$ \( ( 1 - T )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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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

−10.66610716362622758982462772614, −10.23071667737376583706159872533, −9.851895680173775052036593960974, −9.147084726437311251292084534402, −8.997557921523313180826912865509, −8.549687894464285005380232598372, −8.362712811836855626608715311180, −7.41566701458488614582239574045, −7.35848124263822718740617964661, −6.66968579339465231041763409932, −6.47753792155094213698673409059, −5.75982551728490804295116825813, −5.18191648475780262883773136600, −5.07739709847711050946444095747, −4.41192657316577392882689420804, −3.52770520345652397958764540371, −3.43420099191489485531905242408, −2.43388220621920797108667822332, −2.22687602096433717607873770995, −0.968622543135008780305757496039, 0.968622543135008780305757496039, 2.22687602096433717607873770995, 2.43388220621920797108667822332, 3.43420099191489485531905242408, 3.52770520345652397958764540371, 4.41192657316577392882689420804, 5.07739709847711050946444095747, 5.18191648475780262883773136600, 5.75982551728490804295116825813, 6.47753792155094213698673409059, 6.66968579339465231041763409932, 7.35848124263822718740617964661, 7.41566701458488614582239574045, 8.362712811836855626608715311180, 8.549687894464285005380232598372, 8.997557921523313180826912865509, 9.147084726437311251292084534402, 9.851895680173775052036593960974, 10.23071667737376583706159872533, 10.66610716362622758982462772614

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