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Properties

Label 4-275e2-1.1-c1e2-0-2
Degree $4$
Conductor $75625$
Sign $1$
Analytic cond. $4.82191$
Root an. cond. $1.48185$
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  − 2·2-s + 4-s + 4·7-s + 2·9-s + 2·11-s + 8·13-s − 8·14-s + 16-s − 8·17-s − 4·18-s − 4·22-s − 16·26-s + 4·28-s + 4·29-s + 2·32-s + 16·34-s + 2·36-s + 4·37-s + 12·41-s + 12·43-s + 2·44-s − 2·49-s + 8·52-s − 12·53-s − 8·58-s − 8·59-s + 4·61-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 1.51·7-s + 2/3·9-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 − 0.852·22-s − 3.13·26-s + 0.755·28-s + 0.742·29-s + 0.353·32-s + 2.74·34-s + 1/3·36-s + 0.657·37-s + 1.87·41-s + 1.82·43-s + 0.301·44-s − 2/7·49-s + 1.10·52-s − 1.64·53-s − 1.05·58-s − 1.04·59-s + 0.512·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(75625\)    =    \(5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(4.82191\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 75625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8522848639\)
\(L(\frac12)\) \(\approx\) \(0.8522848639\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.2.c_d
3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
13$D_{4}$ \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.13.ai_bi
17$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bq
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.23.a_bm
29$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_be
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_bu
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.43.am_es
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.47.a_di
53$D_{4}$ \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_eg
59$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_dy
61$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_ac
67$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_da
71$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.71.a_o
73$D_{4}$ \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_fy
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.89.e_cc
97$D_{4}$ \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_gk
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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

−11.85078764068031628780781210826, −11.34152724924583538335689542679, −11.06226320084177966638189511932, −10.76332109077944369674429873619, −10.34216843190214001821092395945, −9.364991687812248303675834307232, −9.145660341697595489858477378965, −9.030176806043449037762894238010, −8.181288028444606161175625940356, −8.079260243639446110251027998211, −7.63373519091757555317365298931, −6.55117649324971760946554101508, −6.48170268766400170628637889021, −5.78170041068762991597464841365, −4.79496409631693048558279083330, −4.32093856880512229575824594631, −3.91275786896581322341708444462, −2.68087087515998116401091200236, −1.62076057261134817791620072556, −1.05348097827727392870510429459, 1.05348097827727392870510429459, 1.62076057261134817791620072556, 2.68087087515998116401091200236, 3.91275786896581322341708444462, 4.32093856880512229575824594631, 4.79496409631693048558279083330, 5.78170041068762991597464841365, 6.48170268766400170628637889021, 6.55117649324971760946554101508, 7.63373519091757555317365298931, 8.079260243639446110251027998211, 8.181288028444606161175625940356, 9.030176806043449037762894238010, 9.145660341697595489858477378965, 9.364991687812248303675834307232, 10.34216843190214001821092395945, 10.76332109077944369674429873619, 11.06226320084177966638189511932, 11.34152724924583538335689542679, 11.85078764068031628780781210826

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