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Properties

Label 4-850e2-1.1-c1e2-0-29
Degree $4$
Conductor $722500$
Sign $1$
Analytic cond. $46.0672$
Root an. cond. $2.60524$
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 + 2·3-s + 3·4-s + 4·6-s + 2·7-s + 4·8-s − 9-s + 4·11-s + 6·12-s + 2·13-s + 4·14-s + 5·16-s + 2·17-s − 2·18-s − 4·19-s + 4·21-s + 8·22-s + 4·23-s + 8·24-s + 4·26-s − 6·27-s + 6·28-s − 8·29-s + 6·31-s + 6·32-s + 8·33-s + 4·34-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s + 0.755·7-s + 1.41·8-s − 1/3·9-s + 1.20·11-s + 1.73·12-s + 0.554·13-s + 1.06·14-s + 5/4·16-s + 0.485·17-s − 0.471·18-s − 0.917·19-s + 0.872·21-s + 1.70·22-s + 0.834·23-s + 1.63·24-s + 0.784·26-s − 1.15·27-s + 1.13·28-s − 1.48·29-s + 1.07·31-s + 1.06·32-s + 1.39·33-s + 0.685·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(722500\)    =    \(2^{2} \cdot 5^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(46.0672\)
Root analytic conductor: \(2.60524\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 722500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.973810543\)
\(L(\frac12)\) \(\approx\) \(8.973810543\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad2$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
17$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_f
7$D_{4}$ \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_n
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$D_{4}$ \( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_af
19$D_{4}$ \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_bi
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_s
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.29.i_cw
31$D_{4}$ \( 1 - 6 T + 69 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_cr
37$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.37.a_bq
41$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_cc
43$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.43.a_abq
47$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_es
53$D_{4}$ \( 1 - 6 T + 107 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_ed
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.59.q_ha
61$D_{4}$ \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_dq
67$D_{4}$ \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_w
71$D_{4}$ \( 1 + 18 T + 205 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.71.s_hx
73$D_{4}$ \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_fa
79$D_{4}$ \( 1 + 6 T + 149 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.79.g_ft
83$D_{4}$ \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.83.aq_hq
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.89.e_ha
97$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.97.a_hm
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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

−10.47069240927478387799763706944, −10.10640869941663411220091984026, −9.242612486354718357815062439777, −9.198514486617210582380378707185, −8.589187480512094337243592665666, −8.381270529486954983813656138567, −7.71457603136798102201009236412, −7.49363843435543192554939825479, −6.91343629717678556288113848850, −6.36060518200143086884817831643, −5.93135716561353587294568348125, −5.66184430493732938704079295842, −4.88580149021702740355794221316, −4.54314208037134971567913950026, −3.84239934552070872722459037907, −3.72195653013863120453022588324, −2.94480137474653486855259468425, −2.61350944777162393375297656071, −1.85257139337024660809808365529, −1.25640618338636972041896897550, 1.25640618338636972041896897550, 1.85257139337024660809808365529, 2.61350944777162393375297656071, 2.94480137474653486855259468425, 3.72195653013863120453022588324, 3.84239934552070872722459037907, 4.54314208037134971567913950026, 4.88580149021702740355794221316, 5.66184430493732938704079295842, 5.93135716561353587294568348125, 6.36060518200143086884817831643, 6.91343629717678556288113848850, 7.49363843435543192554939825479, 7.71457603136798102201009236412, 8.381270529486954983813656138567, 8.589187480512094337243592665666, 9.198514486617210582380378707185, 9.242612486354718357815062439777, 10.10640869941663411220091984026, 10.47069240927478387799763706944

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