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Properties

Label 4-2925e2-1.1-c0e2-0-1
Degree $4$
Conductor $8555625$
Sign $1$
Analytic cond. $2.13091$
Root an. cond. $1.20820$
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  + 2·7-s − 2·13-s − 16-s + 2·19-s + 2·31-s − 2·37-s + 2·49-s − 2·67-s − 2·73-s − 4·91-s + 2·97-s + 2·109-s − 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 2·7-s − 2·13-s − 16-s + 2·19-s + 2·31-s − 2·37-s + 2·49-s − 2·67-s − 2·73-s − 4·91-s + 2·97-s + 2·109-s − 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8555625\)    =    \(3^{4} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2.13091\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8555625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.593637127\)
\(L(\frac12)\) \(\approx\) \(1.593637127\)
\(L(1)\) 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 \( 1 \)
5 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
11$C_2^2$ \( 1 + T^{4} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2^2$ \( 1 + T^{4} \)
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

−8.877432391173805065918573551583, −8.796565446771411618030449092970, −8.452107150053014218798403494712, −7.83694491685249558842204458585, −7.63053142933933752874481358603, −7.23841852989063778615383911110, −7.19895575417817371103485691333, −6.55084536189553412616567618787, −6.03038001926911147283394911743, −5.49477803590909605609037465802, −5.17729934087253494757197033670, −4.81529581128304795641842446276, −4.57442749646993365564012442219, −4.33891591300115550590024760661, −3.47264431108151420695899848668, −2.94486511101192324381991584207, −2.63962268217671986742734840564, −1.83525056590084789969874532424, −1.73272705387019840550746620817, −0.813072678540964668806025214170, 0.813072678540964668806025214170, 1.73272705387019840550746620817, 1.83525056590084789969874532424, 2.63962268217671986742734840564, 2.94486511101192324381991584207, 3.47264431108151420695899848668, 4.33891591300115550590024760661, 4.57442749646993365564012442219, 4.81529581128304795641842446276, 5.17729934087253494757197033670, 5.49477803590909605609037465802, 6.03038001926911147283394911743, 6.55084536189553412616567618787, 7.19895575417817371103485691333, 7.23841852989063778615383911110, 7.63053142933933752874481358603, 7.83694491685249558842204458585, 8.452107150053014218798403494712, 8.796565446771411618030449092970, 8.877432391173805065918573551583

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