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Properties

Label 4-750e2-1.1-c1e2-0-3
Degree $4$
Conductor $562500$
Sign $1$
Analytic cond. $35.8654$
Root an. cond. $2.44719$
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  − 2·2-s + 2·3-s + 3·4-s − 4·6-s + 7·7-s − 4·8-s + 3·9-s − 5·11-s + 6·12-s + 9·13-s − 14·14-s + 5·16-s + 2·17-s − 6·18-s − 3·19-s + 14·21-s + 10·22-s + 5·23-s − 8·24-s − 18·26-s + 4·27-s + 21·28-s − 2·29-s − 4·31-s − 6·32-s − 10·33-s − 4·34-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s + 2.64·7-s − 1.41·8-s + 9-s − 1.50·11-s + 1.73·12-s + 2.49·13-s − 3.74·14-s + 5/4·16-s + 0.485·17-s − 1.41·18-s − 0.688·19-s + 3.05·21-s + 2.13·22-s + 1.04·23-s − 1.63·24-s − 3.53·26-s + 0.769·27-s + 3.96·28-s − 0.371·29-s − 0.718·31-s − 1.06·32-s − 1.74·33-s − 0.685·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 562500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 562500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(562500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(35.8654\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 562500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.527940844\)
\(L(\frac12)\) \(\approx\) \(2.527940844\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
good7$D_{4}$ \( 1 - p T + 25 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.7.ah_z
11$D_{4}$ \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.11.f_r
13$D_{4}$ \( 1 - 9 T + 45 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.13.aj_bt
17$D_{4}$ \( 1 - 2 T - 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_ak
19$D_{4}$ \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.19.d_j
23$D_{4}$ \( 1 - 5 T + 21 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.23.af_v
29$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_o
31$C_4$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_bu
37$D_{4}$ \( 1 + T + 73 T^{2} + p T^{3} + p^{2} T^{4} \) 2.37.b_cv
41$D_{4}$ \( 1 - 3 T + 83 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_df
43$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.43.ac_bq
47$D_{4}$ \( 1 - 15 T + 149 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.47.ap_ft
53$D_{4}$ \( 1 - 3 T + 107 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_ed
59$D_{4}$ \( 1 + 15 T + 163 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.59.p_gh
61$D_{4}$ \( 1 - 10 T + 102 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.61.ak_dy
67$D_{4}$ \( 1 - 2 T + 130 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_fa
71$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \) 2.71.a_es
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.73.q_ic
79$D_{4}$ \( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.79.aq_hu
83$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.83.ae_go
89$D_{4}$ \( 1 - 11 T + 107 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.89.al_ed
97$D_{4}$ \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_da
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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

−10.60117460001814790123815808961, −10.36159654423022920317707907717, −9.493608215871121679620853072776, −8.972527151304895505297455197288, −8.808798108697884325141960292324, −8.461571378487726807263584562430, −8.043164029822116798756954868928, −7.84462332378714446137790700393, −7.42870943787904088195404995620, −7.09696044061933819000435744988, −6.13507820152330042751533510841, −5.84832147928909660512638598894, −5.10886372119344622084759441837, −4.79444024909693425982103301951, −3.87869696057505651806647739525, −3.59445857792095036750700771086, −2.54159153840049157841681421308, −2.29213383794753257312806161143, −1.36708512983003916590438478009, −1.21254035295304341454546610332, 1.21254035295304341454546610332, 1.36708512983003916590438478009, 2.29213383794753257312806161143, 2.54159153840049157841681421308, 3.59445857792095036750700771086, 3.87869696057505651806647739525, 4.79444024909693425982103301951, 5.10886372119344622084759441837, 5.84832147928909660512638598894, 6.13507820152330042751533510841, 7.09696044061933819000435744988, 7.42870943787904088195404995620, 7.84462332378714446137790700393, 8.043164029822116798756954868928, 8.461571378487726807263584562430, 8.808798108697884325141960292324, 8.972527151304895505297455197288, 9.493608215871121679620853072776, 10.36159654423022920317707907717, 10.60117460001814790123815808961

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