Properties

Label 4-475e2-1.1-c1e2-0-2
Degree $4$
Conductor $225625$
Sign $1$
Analytic cond. $14.3860$
Root an. cond. $1.94753$
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  + 2·2-s + 2·4-s + 8·7-s + 4·8-s + 3·9-s − 2·11-s − 2·13-s + 16·14-s + 8·16-s − 2·17-s + 6·18-s − 7·19-s − 4·22-s + 6·23-s − 4·26-s + 16·28-s − 9·29-s − 14·31-s + 8·32-s − 4·34-s + 6·36-s − 4·37-s − 14·38-s − 2·41-s + 2·43-s − 4·44-s + 12·46-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 3.02·7-s + 1.41·8-s + 9-s − 0.603·11-s − 0.554·13-s + 4.27·14-s + 2·16-s − 0.485·17-s + 1.41·18-s − 1.60·19-s − 0.852·22-s + 1.25·23-s − 0.784·26-s + 3.02·28-s − 1.67·29-s − 2.51·31-s + 1.41·32-s − 0.685·34-s + 36-s − 0.657·37-s − 2.27·38-s − 0.312·41-s + 0.304·43-s − 0.603·44-s + 1.76·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(225625\)    =    \(5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(14.3860\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 225625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.612719751\)
\(L(\frac12)\) \(\approx\) \(5.612719751\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
19$C_2$ \( 1 + 7 T + p T^{2} \)
good2$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + T - 70 T^{2} + p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 11 T + 32 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 6 T - 61 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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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

−11.21206814392667914991093789148, −10.90471578686655770881722212476, −10.47932195585320547667850945242, −10.47926428389554250498594186373, −9.231403414712642924893437783538, −9.066647111327061047407457202854, −8.284883554916736610418978676258, −7.935791464766070634966570650783, −7.43139367874779369906396395452, −7.28682255474782080712788413973, −6.70759651690475235308707442941, −5.53545734999154402714993023198, −5.47336379223912505359138887712, −4.80195433734292026316143436394, −4.75128138598538874450553548668, −3.98980113475805215523919362745, −3.85089679886022452512250861402, −2.45324272264237617901690965065, −1.77629153868512302240926579949, −1.60205579459502770893204042843, 1.60205579459502770893204042843, 1.77629153868512302240926579949, 2.45324272264237617901690965065, 3.85089679886022452512250861402, 3.98980113475805215523919362745, 4.75128138598538874450553548668, 4.80195433734292026316143436394, 5.47336379223912505359138887712, 5.53545734999154402714993023198, 6.70759651690475235308707442941, 7.28682255474782080712788413973, 7.43139367874779369906396395452, 7.935791464766070634966570650783, 8.284883554916736610418978676258, 9.066647111327061047407457202854, 9.231403414712642924893437783538, 10.47926428389554250498594186373, 10.47932195585320547667850945242, 10.90471578686655770881722212476, 11.21206814392667914991093789148

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