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Properties

Label 4-875e2-1.1-c1e2-0-2
Degree $4$
Conductor $765625$
Sign $1$
Analytic cond. $48.8169$
Root an. cond. $2.64327$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s − 2·4-s − 2·6-s + 2·7-s − 3·8-s − 3·9-s − 11-s + 4·12-s − 2·13-s + 2·14-s + 16-s + 17-s − 3·18-s − 5·19-s − 4·21-s − 22-s + 3·23-s + 6·24-s − 2·26-s + 14·27-s − 4·28-s − 10·29-s − 11·31-s + 2·32-s + 2·33-s + 34-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 4-s − 0.816·6-s + 0.755·7-s − 1.06·8-s − 9-s − 0.301·11-s + 1.15·12-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.707·18-s − 1.14·19-s − 0.872·21-s − 0.213·22-s + 0.625·23-s + 1.22·24-s − 0.392·26-s + 2.69·27-s − 0.755·28-s − 1.85·29-s − 1.97·31-s + 0.353·32-s + 0.348·33-s + 0.171·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 765625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(765625\)    =    \(5^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(48.8169\)
Root analytic conductor: \(2.64327\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 765625,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_d
3$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.3.c_h
11$C_4$ \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) 2.11.b_v
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_w
17$D_{4}$ \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) 2.17.ab_bh
19$D_{4}$ \( 1 + 5 T + 43 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.19.f_br
23$D_{4}$ \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_bv
29$D_{4}$ \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.29.k_cl
31$C_4$ \( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.31.l_dd
37$D_{4}$ \( 1 - 11 T + 103 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.37.al_dz
41$D_{4}$ \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_bu
43$D_{4}$ \( 1 + 7 T + 97 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.43.h_dt
47$D_{4}$ \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.47.o_fi
53$D_{4}$ \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.53.c_bb
59$D_{4}$ \( 1 + 15 T + 173 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.59.p_gr
61$D_{4}$ \( 1 + 21 T + 221 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.61.v_in
67$D_{4}$ \( 1 + 4 T + 93 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_dp
71$D_{4}$ \( 1 - 14 T + 111 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.71.ao_eh
73$D_{4}$ \( 1 + 7 T + 57 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.73.h_cf
79$D_{4}$ \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_fi
83$D_{4}$ \( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.83.as_it
89$C_2^2$ \( 1 + 173 T^{2} + p^{2} T^{4} \) 2.89.a_gr
97$D_{4}$ \( 1 + 4 T + 178 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_gw
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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

−9.857388457070266898888667174282, −9.504293091970184072616920160264, −9.033549287240256418415968703809, −8.833637306600802759746640803194, −8.107251406229462053616382725200, −8.003357629916469990398266204438, −7.36959319037194757268846466791, −6.78559899120308481029968847396, −6.06366748124968947208499508610, −5.99485548263870368897867466847, −5.36004350022813211029104726733, −5.11625073559353704935616401009, −4.64107446874723392284030441610, −4.44172549613830580121478771189, −3.37442471178928085032651804680, −3.32429406526622042849310199714, −2.30871070290266741892515366941, −1.54253429324590232503219253634, 0, 0, 1.54253429324590232503219253634, 2.30871070290266741892515366941, 3.32429406526622042849310199714, 3.37442471178928085032651804680, 4.44172549613830580121478771189, 4.64107446874723392284030441610, 5.11625073559353704935616401009, 5.36004350022813211029104726733, 5.99485548263870368897867466847, 6.06366748124968947208499508610, 6.78559899120308481029968847396, 7.36959319037194757268846466791, 8.003357629916469990398266204438, 8.107251406229462053616382725200, 8.833637306600802759746640803194, 9.033549287240256418415968703809, 9.504293091970184072616920160264, 9.857388457070266898888667174282

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