Properties

Label 4-65e2-1.1-c1e2-0-0
Degree $4$
Conductor $4225$
Sign $1$
Analytic cond. $0.269389$
Root an. cond. $0.720435$
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 − 4-s − 2·5-s + 8·8-s + 2·9-s + 4·10-s + 6·13-s − 7·16-s − 4·18-s + 2·20-s − 25-s − 12·26-s + 12·29-s − 14·32-s − 2·36-s + 12·37-s − 16·40-s − 4·45-s − 16·47-s − 14·49-s + 2·50-s − 6·52-s − 24·58-s + 12·61-s + 35·64-s − 12·65-s + 24·67-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s − 0.894·5-s + 2.82·8-s + 2/3·9-s + 1.26·10-s + 1.66·13-s − 7/4·16-s − 0.942·18-s + 0.447·20-s − 1/5·25-s − 2.35·26-s + 2.22·29-s − 2.47·32-s − 1/3·36-s + 1.97·37-s − 2.52·40-s − 0.596·45-s − 2.33·47-s − 2·49-s + 0.282·50-s − 0.832·52-s − 3.15·58-s + 1.53·61-s + 35/8·64-s − 1.48·65-s + 2.93·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.269389\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{65} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3054496044\)
\(L(\frac12)\) \(\approx\) \(0.3054496044\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 138 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−15.48094517667249786984419820405, −14.35357484071792060463199967707, −14.30599507841215133090697000144, −13.29225623876548200001606745781, −13.13503890469136268151357292289, −12.58532070036526148632685119076, −11.35492077341153365777874748609, −11.32139467924123906337733892314, −10.43059520603588603966913018618, −9.764858650237680717334496654142, −9.641698746713342675298466711552, −8.451466216636959955255634250710, −8.403686816275583015400740968023, −7.955940430465568710907714334923, −7.07329553608668792226551336916, −6.26448749475629147424939311323, −4.93630899930433513855533732459, −4.33689139383410961559481614521, −3.60296988112209500923280590170, −1.18396783807390036137058712068, 1.18396783807390036137058712068, 3.60296988112209500923280590170, 4.33689139383410961559481614521, 4.93630899930433513855533732459, 6.26448749475629147424939311323, 7.07329553608668792226551336916, 7.955940430465568710907714334923, 8.403686816275583015400740968023, 8.451466216636959955255634250710, 9.641698746713342675298466711552, 9.764858650237680717334496654142, 10.43059520603588603966913018618, 11.32139467924123906337733892314, 11.35492077341153365777874748609, 12.58532070036526148632685119076, 13.13503890469136268151357292289, 13.29225623876548200001606745781, 14.30599507841215133090697000144, 14.35357484071792060463199967707, 15.48094517667249786984419820405

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