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Properties

Label 4-35e2-1.1-c1e2-0-0
Degree $4$
Conductor $1225$
Sign $1$
Analytic cond. $0.0781070$
Root an. cond. $0.528655$
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-s − 3-s + 4-s + 2·5-s + 6-s − 2·7-s − 3·8-s − 9-s − 2·10-s + 11-s − 12-s + 5·13-s + 2·14-s − 2·15-s + 16-s − 5·17-s + 18-s − 6·19-s + 2·20-s + 2·21-s − 22-s − 2·23-s + 3·24-s + 3·25-s − 5·26-s − 2·28-s + 29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.755·7-s − 1.06·8-s − 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s + 1.38·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s − 1.21·17-s + 0.235·18-s − 1.37·19-s + 0.447·20-s + 0.436·21-s − 0.213·22-s − 0.417·23-s + 0.612·24-s + 3/5·25-s − 0.980·26-s − 0.377·28-s + 0.185·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.0781070\)
Root analytic conductor: \(0.528655\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3727467837\)
\(L(\frac12)\) \(\approx\) \(0.3727467837\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 6 T - 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 9 T + 174 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 6 T + 170 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 9 T + 108 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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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

−16.90322846331634953577134877295, −16.54043015310824634377959576393, −15.74352143908137457099974538480, −15.47778975023287695392669568729, −14.58335792317703476215761000619, −13.94588673316197709173832397868, −12.99212700705068834617634779508, −12.96793398602071777540469125017, −11.90227741643907382762501423525, −11.12711409504629322506872055031, −10.91335028499141244772470576035, −9.982496960603323626825608149426, −9.212565164174387439717095614639, −8.914097901089898875492127200173, −8.054746333088715318280391070413, −6.48150256884439156034400207857, −6.44771243960593321795590797896, −5.75333263909241173971323500780, −4.15686477265978088320503087188, −2.53657717653485394345612541605, 2.53657717653485394345612541605, 4.15686477265978088320503087188, 5.75333263909241173971323500780, 6.44771243960593321795590797896, 6.48150256884439156034400207857, 8.054746333088715318280391070413, 8.914097901089898875492127200173, 9.212565164174387439717095614639, 9.982496960603323626825608149426, 10.91335028499141244772470576035, 11.12711409504629322506872055031, 11.90227741643907382762501423525, 12.96793398602071777540469125017, 12.99212700705068834617634779508, 13.94588673316197709173832397868, 14.58335792317703476215761000619, 15.47778975023287695392669568729, 15.74352143908137457099974538480, 16.54043015310824634377959576393, 16.90322846331634953577134877295

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