Properties

Label 4-605e2-1.1-c1e2-0-0
Degree $4$
Conductor $366025$
Sign $1$
Analytic cond. $23.3380$
Root an. cond. $2.19794$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4-s − 2·5-s + 4·7-s + 2·9-s + 4·10-s + 8·13-s − 8·14-s + 16-s − 8·17-s − 4·18-s − 2·20-s + 3·25-s − 16·26-s + 4·28-s − 4·29-s + 2·32-s + 16·34-s − 8·35-s + 2·36-s − 4·37-s − 12·41-s + 12·43-s − 4·45-s − 2·49-s − 6·50-s + 8·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s − 0.894·5-s + 1.51·7-s + 2/3·9-s + 1.26·10-s + 2.21·13-s − 2.13·14-s + 1/4·16-s − 1.94·17-s − 0.942·18-s − 0.447·20-s + 3/5·25-s − 3.13·26-s + 0.755·28-s − 0.742·29-s + 0.353·32-s + 2.74·34-s − 1.35·35-s + 1/3·36-s − 0.657·37-s − 1.87·41-s + 1.82·43-s − 0.596·45-s − 2/7·49-s − 0.848·50-s + 1.10·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 366025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(366025\)    =    \(5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(23.3380\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{605} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 366025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7053008647\)
\(L(\frac12)\) \(\approx\) \(0.7053008647\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−10.69392841136218057500068232333, −10.61217710221480625620847590002, −10.01616586818574353301921913163, −9.313556481174185678920721122062, −8.870948390669154699685197594437, −8.818366946591205571871160512807, −8.186821795467132538719611022776, −8.176291199733775922108628626945, −7.54090663688034373695167746890, −7.03664113219270685312609507394, −6.55607289928911078161638991098, −6.05427308819497397235336010157, −5.33960226149944408115322300453, −4.70791620867604202670106942539, −4.26236798415421457702648897318, −3.85939434191577354164866479839, −3.16511900956594915660339762596, −2.00202795573304983660140301308, −1.52320903738309950873819603485, −0.64500020425035326738338827898, 0.64500020425035326738338827898, 1.52320903738309950873819603485, 2.00202795573304983660140301308, 3.16511900956594915660339762596, 3.85939434191577354164866479839, 4.26236798415421457702648897318, 4.70791620867604202670106942539, 5.33960226149944408115322300453, 6.05427308819497397235336010157, 6.55607289928911078161638991098, 7.03664113219270685312609507394, 7.54090663688034373695167746890, 8.176291199733775922108628626945, 8.186821795467132538719611022776, 8.818366946591205571871160512807, 8.870948390669154699685197594437, 9.313556481174185678920721122062, 10.01616586818574353301921913163, 10.61217710221480625620847590002, 10.69392841136218057500068232333

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