Properties

Label 4-1805e2-1.1-c1e2-0-4
Degree $4$
Conductor $3258025$
Sign $1$
Analytic cond. $207.734$
Root an. cond. $3.79644$
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  + 4·4-s − 5-s + 6·9-s + 10·11-s + 12·16-s − 4·20-s − 4·25-s + 24·36-s + 40·44-s − 6·45-s − 5·49-s − 10·55-s − 30·61-s + 32·64-s − 12·80-s + 27·81-s + 60·99-s − 16·100-s − 20·101-s + 53·121-s + 9·125-s + 127-s + 131-s + 137-s + 139-s + 72·144-s + 149-s + ⋯
L(s)  = 1  + 2·4-s − 0.447·5-s + 2·9-s + 3.01·11-s + 3·16-s − 0.894·20-s − 4/5·25-s + 4·36-s + 6.03·44-s − 0.894·45-s − 5/7·49-s − 1.34·55-s − 3.84·61-s + 4·64-s − 1.34·80-s + 3·81-s + 6.03·99-s − 8/5·100-s − 1.99·101-s + 4.81·121-s + 0.804·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 6·144-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3258025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3258025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3258025\)    =    \(5^{2} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(207.734\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1805} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3258025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.613379386\)
\(L(\frac12)\) \(\approx\) \(6.613379386\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
19 \( 1 \)
good2$C_2$ \( ( 1 - p T^{2} )^{2} \)
3$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - p T^{2} )^{2} \)
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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

−9.436385365048427922057813788004, −9.420264481221764838487302267372, −8.654291319180622309730142326698, −8.066024701715080433411156361123, −7.79673289714273862146197698551, −7.32784909456880983184783824028, −7.01185268922805688902751875529, −6.77199392012558742286058165180, −6.44590779767483461174758937445, −6.04557888591264506973551174429, −5.75431925473642207948907103346, −4.81898104525831975047546655384, −4.33917776647577494876924816201, −4.07479631525240120646021361236, −3.41206874872959085804217333513, −3.37818609089530100132707616139, −2.40443777649611652640449082062, −1.67373409149660737061842670940, −1.52684418489301665659941999425, −1.09138253188100489468037931358, 1.09138253188100489468037931358, 1.52684418489301665659941999425, 1.67373409149660737061842670940, 2.40443777649611652640449082062, 3.37818609089530100132707616139, 3.41206874872959085804217333513, 4.07479631525240120646021361236, 4.33917776647577494876924816201, 4.81898104525831975047546655384, 5.75431925473642207948907103346, 6.04557888591264506973551174429, 6.44590779767483461174758937445, 6.77199392012558742286058165180, 7.01185268922805688902751875529, 7.32784909456880983184783824028, 7.79673289714273862146197698551, 8.066024701715080433411156361123, 8.654291319180622309730142326698, 9.420264481221764838487302267372, 9.436385365048427922057813788004

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