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Properties

Label 4-55e2-1.1-c1e2-0-1
Degree $4$
Conductor $3025$
Sign $1$
Analytic cond. $0.192876$
Root an. cond. $0.662704$
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 + 2·11-s − 8·13-s − 8·14-s + 16-s + 8·17-s + 4·18-s − 2·20-s + 4·22-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 + 2·44-s − 4·45-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 + 0.603·11-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 + 0.852·22-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 + 0.301·44-s − 0.596·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.192876\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.065356079\)
\(L(\frac12)\) \(\approx\) \(1.065356079\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
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

−15.34960948708537437517013199535, −14.94939385721676879893707388798, −14.35642538898121481045712364568, −14.10678218029280958479780612667, −13.25975961075160009500758146644, −12.69871244487788176118942150891, −12.32439160511051109383228422372, −12.18106320759827562104741174082, −11.34576766095860460302436742769, −10.09200070663283635358165303574, −10.01954456510364785732734865775, −9.373862471307683607328605854679, −8.288711442380031784730947830553, −7.24885125285007192371091012034, −7.19557878469241887119343312891, −6.08854885975527628228900382813, −5.16733014703725630411671370919, −4.53592171229370235630296301299, −3.70347734990671907639955801345, −3.03212513583617232663088011462, 3.03212513583617232663088011462, 3.70347734990671907639955801345, 4.53592171229370235630296301299, 5.16733014703725630411671370919, 6.08854885975527628228900382813, 7.19557878469241887119343312891, 7.24885125285007192371091012034, 8.288711442380031784730947830553, 9.373862471307683607328605854679, 10.01954456510364785732734865775, 10.09200070663283635358165303574, 11.34576766095860460302436742769, 12.18106320759827562104741174082, 12.32439160511051109383228422372, 12.69871244487788176118942150891, 13.25975961075160009500758146644, 14.10678218029280958479780612667, 14.35642538898121481045712364568, 14.94939385721676879893707388798, 15.34960948708537437517013199535

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