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Properties

Label 4-1040e2-1.1-c1e2-0-52
Degree $4$
Conductor $1081600$
Sign $1$
Analytic cond. $68.9637$
Root an. cond. $2.88174$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·5-s + 6·11-s + 2·13-s − 4·15-s + 10·19-s + 6·23-s + 3·25-s − 2·27-s − 4·29-s + 6·31-s + 12·33-s + 8·37-s + 4·39-s − 16·41-s + 2·43-s + 16·47-s − 2·49-s + 16·53-s − 12·55-s + 20·57-s + 14·59-s − 4·61-s − 4·65-s − 4·67-s + 12·69-s − 6·71-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 1.80·11-s + 0.554·13-s − 1.03·15-s + 2.29·19-s + 1.25·23-s + 3/5·25-s − 0.384·27-s − 0.742·29-s + 1.07·31-s + 2.08·33-s + 1.31·37-s + 0.640·39-s − 2.49·41-s + 0.304·43-s + 2.33·47-s − 2/7·49-s + 2.19·53-s − 1.61·55-s + 2.64·57-s + 1.82·59-s − 0.512·61-s − 0.496·65-s − 0.488·67-s + 1.44·69-s − 0.712·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1081600\)    =    \(2^{8} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(68.9637\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1081600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.621562708\)
\(L(\frac12)\) \(\approx\) \(3.621562708\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_e
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_bc
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.17.a_w
19$D_{4}$ \( 1 - 10 T + 60 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.19.ak_ci
23$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_bc
29$D_{4}$ \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_by
31$D_{4}$ \( 1 - 6 T + 68 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_cq
37$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_bq
41$D_{4}$ \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.41.q_fe
43$D_{4}$ \( 1 - 2 T + 84 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.43.ac_dg
47$D_{4}$ \( 1 - 16 T + 146 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.47.aq_fq
53$D_{4}$ \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.53.aq_gc
59$D_{4}$ \( 1 - 14 T + 140 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.59.ao_fk
61$D_{4}$ \( 1 + 4 T + 114 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_ek
67$D_{4}$ \( 1 + 4 T - 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_acc
71$D_{4}$ \( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_cy
73$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.73.a_fq
79$D_{4}$ \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_ha
83$D_{4}$ \( 1 + 16 T + 218 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.83.q_ik
89$D_{4}$ \( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.89.ae_fe
97$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_g
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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

−9.999379055087354470562182893044, −9.607289652788812170525813313921, −9.061387198178810891905373000907, −8.924555022196628011050957817107, −8.533282882079795593937326184557, −8.267917722343081265173796207555, −7.54007788811215094041258116486, −7.31366823778085007945657570058, −6.95971742468071165864179048534, −6.50762305406485901185523222796, −5.73974034555109740290873394545, −5.51871229711353571803336379500, −4.79262259341883547799046643314, −4.20298920860596348802360263558, −3.79407174757904391751139668913, −3.40981136754528865536639413729, −2.93705634094810328189287060876, −2.46232651342925130187797079509, −1.29342481061562879556021312902, −0.983973771721332173838433557496, 0.983973771721332173838433557496, 1.29342481061562879556021312902, 2.46232651342925130187797079509, 2.93705634094810328189287060876, 3.40981136754528865536639413729, 3.79407174757904391751139668913, 4.20298920860596348802360263558, 4.79262259341883547799046643314, 5.51871229711353571803336379500, 5.73974034555109740290873394545, 6.50762305406485901185523222796, 6.95971742468071165864179048534, 7.31366823778085007945657570058, 7.54007788811215094041258116486, 8.267917722343081265173796207555, 8.533282882079795593937326184557, 8.924555022196628011050957817107, 9.061387198178810891905373000907, 9.607289652788812170525813313921, 9.999379055087354470562182893044

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