Properties

Label 4-525e2-1.1-c1e2-0-0
Degree $4$
Conductor $275625$
Sign $1$
Analytic cond. $17.5740$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 2·4-s − 4·7-s + 6·9-s + 6·12-s + 6·19-s + 12·21-s − 9·27-s + 8·28-s − 15·31-s − 12·36-s − 11·37-s + 10·43-s + 9·49-s − 18·57-s + 27·61-s − 24·63-s + 8·64-s − 16·67-s + 3·73-s − 12·76-s − 17·79-s + 9·81-s − 24·84-s + 45·93-s − 27·103-s + 18·108-s + ⋯
L(s)  = 1  − 1.73·3-s − 4-s − 1.51·7-s + 2·9-s + 1.73·12-s + 1.37·19-s + 2.61·21-s − 1.73·27-s + 1.51·28-s − 2.69·31-s − 2·36-s − 1.80·37-s + 1.52·43-s + 9/7·49-s − 2.38·57-s + 3.45·61-s − 3.02·63-s + 64-s − 1.95·67-s + 0.351·73-s − 1.37·76-s − 1.91·79-s + 81-s − 2.61·84-s + 4.66·93-s − 2.66·103-s + 1.73·108-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(275625\)    =    \(3^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(17.5740\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 275625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2191689427\)
\(L(\frac12)\) \(\approx\) \(0.2191689427\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 + p T + p T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.98388937734398707745362662684, −10.70502872866695506007782725938, −10.16439879762252641259284961504, −9.735795588185913057371186093046, −9.549037082288771884611042367968, −8.875271833987165040728417833889, −8.773706315617616460455465209010, −7.75869065758754155126310766525, −7.11471536086813389371964565444, −7.08010138820884867212818598528, −6.49284029989436373858784559154, −5.77625279446492408405608457716, −5.48896659900045136605971722695, −5.24466650318102254395579568711, −4.51713878205467890193379286635, −3.72688113226704224038148664973, −3.69431087386852630741316490631, −2.60421819551448974455189700880, −1.39515444855475773652534463476, −0.33065262391938944937574170585, 0.33065262391938944937574170585, 1.39515444855475773652534463476, 2.60421819551448974455189700880, 3.69431087386852630741316490631, 3.72688113226704224038148664973, 4.51713878205467890193379286635, 5.24466650318102254395579568711, 5.48896659900045136605971722695, 5.77625279446492408405608457716, 6.49284029989436373858784559154, 7.08010138820884867212818598528, 7.11471536086813389371964565444, 7.75869065758754155126310766525, 8.773706315617616460455465209010, 8.875271833987165040728417833889, 9.549037082288771884611042367968, 9.735795588185913057371186093046, 10.16439879762252641259284961504, 10.70502872866695506007782725938, 10.98388937734398707745362662684

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