Properties

Label 4-11225-1.1-c1e2-0-1
Degree $4$
Conductor $11225$
Sign $-1$
Analytic cond. $0.715715$
Root an. cond. $0.919782$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 2·5-s − 4·9-s − 2·11-s + 4·20-s − 25-s − 14·29-s + 3·31-s + 8·36-s + 4·41-s + 4·44-s + 8·45-s + 5·49-s + 4·55-s − 6·59-s + 9·61-s + 8·64-s − 8·71-s + 9·79-s + 7·81-s + 2·89-s + 8·99-s + 2·100-s + 4·101-s + 109-s + 28·116-s − 15·121-s + ⋯
L(s)  = 1  − 4-s − 0.894·5-s − 4/3·9-s − 0.603·11-s + 0.894·20-s − 1/5·25-s − 2.59·29-s + 0.538·31-s + 4/3·36-s + 0.624·41-s + 0.603·44-s + 1.19·45-s + 5/7·49-s + 0.539·55-s − 0.781·59-s + 1.15·61-s + 64-s − 0.949·71-s + 1.01·79-s + 7/9·81-s + 0.211·89-s + 0.804·99-s + 1/5·100-s + 0.398·101-s + 0.0957·109-s + 2.59·116-s − 1.36·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11225\)    =    \(5^{2} \cdot 449\)
Sign: $-1$
Analytic conductor: \(0.715715\)
Root analytic conductor: \(0.919782\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 11225,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad5$C_2$ \( 1 + 2 T + p T^{2} \)
449$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 6 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.2.a_c
3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.c_t
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.17.a_ae
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.23.a_abj
29$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.o_ec
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.ad_ca
37$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.37.a_aq
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ae_cs
43$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.43.a_aba
47$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.47.a_bc
53$C_2^2$ \( 1 + 39 T^{2} + p^{2} T^{4} \) 2.53.a_bn
59$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.59.g_bb
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.aj_ei
67$C_2^2$ \( 1 + 59 T^{2} + p^{2} T^{4} \) 2.67.a_ch
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.i_fm
73$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.73.a_dc
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.aj_ec
83$C_2^2$ \( 1 - 88 T^{2} + p^{2} T^{4} \) 2.83.a_adk
89$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.89.ac_el
97$C_2^2$ \( 1 + 89 T^{2} + p^{2} T^{4} \) 2.97.a_dl
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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

−11.31521002529604527190696345147, −10.67692895790493761087447254611, −9.984141979258291623851733674030, −9.176590825093271365252734688293, −9.011825810356457251530435608082, −8.272436784497754023219058151390, −7.79562370721089318994581935352, −7.30531580998987159068638276744, −6.25483153408191298867284232361, −5.55281403418143781490173763676, −5.04263241417144358939568037454, −4.11537653907052538468041095390, −3.56654142705823616342891809525, −2.47403788859579255793990179099, 0, 2.47403788859579255793990179099, 3.56654142705823616342891809525, 4.11537653907052538468041095390, 5.04263241417144358939568037454, 5.55281403418143781490173763676, 6.25483153408191298867284232361, 7.30531580998987159068638276744, 7.79562370721089318994581935352, 8.272436784497754023219058151390, 9.011825810356457251530435608082, 9.176590825093271365252734688293, 9.984141979258291623851733674030, 10.67692895790493761087447254611, 11.31521002529604527190696345147

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