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Properties

Label 4-7350e2-1.1-c1e2-0-2
Degree $4$
Conductor $54022500$
Sign $1$
Analytic cond. $3444.52$
Root an. cond. $7.66094$
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 + 4·8-s + 3·9-s − 4·11-s − 6·12-s + 6·13-s + 5·16-s + 2·17-s + 6·18-s + 8·19-s − 8·22-s + 6·23-s − 8·24-s + 12·26-s − 4·27-s − 10·29-s + 6·31-s + 6·32-s + 8·33-s + 4·34-s + 9·36-s − 8·37-s + 16·38-s − 12·39-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.63·6-s + 1.41·8-s + 9-s − 1.20·11-s − 1.73·12-s + 1.66·13-s + 5/4·16-s + 0.485·17-s + 1.41·18-s + 1.83·19-s − 1.70·22-s + 1.25·23-s − 1.63·24-s + 2.35·26-s − 0.769·27-s − 1.85·29-s + 1.07·31-s + 1.06·32-s + 1.39·33-s + 0.685·34-s + 3/2·36-s − 1.31·37-s + 2.59·38-s − 1.92·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(54022500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3444.52\)
Root analytic conductor: \(7.66094\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 54022500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.973565929\)
\(L(\frac12)\) \(\approx\) \(6.973565929\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good11$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_i
13$D_{4}$ \( 1 - 6 T + 27 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_bb
17$D_{4}$ \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_d
19$D_{4}$ \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.19.ai_ca
23$D_{4}$ \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_cb
29$D_{4}$ \( 1 + 10 T + 51 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.29.k_bz
31$D_{4}$ \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_cb
37$D_{4}$ \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_cu
41$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.41.ao_fb
43$D_{4}$ \( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_ef
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$D_{4}$ \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_df
59$D_{4}$ \( 1 + 6 T + 125 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_ev
61$D_{4}$ \( 1 + 6 T + 99 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_dv
67$D_{4}$ \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_eo
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.71.e_fq
73$D_{4}$ \( 1 - 8 T + 144 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_fo
79$D_{4}$ \( 1 - 4 T + 144 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_fo
83$D_{4}$ \( 1 + 14 T + 213 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.83.o_if
89$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.89.ai_co
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_pa
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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

−7.66263692450090686510326776552, −7.65489168011193579071417810725, −7.20631049385956534606659243840, −7.12412960075898860988496092559, −6.35749665209084826093271008217, −6.29731244939544276947778215530, −5.76893529579824842087721529913, −5.64424219206451047630738744027, −5.22018031788748319305832197204, −5.14767662828047935785018067411, −4.51227245967404926376717183911, −4.36113260080479830342227147056, −3.64586933829706876456116285702, −3.47257796576051962867658091826, −3.13772720771022185391582059998, −2.67878774145804997166649562950, −2.05887782358820700963961461289, −1.54741805482697464885848866774, −1.02813631580627332876986671684, −0.61575118696838810207196711915, 0.61575118696838810207196711915, 1.02813631580627332876986671684, 1.54741805482697464885848866774, 2.05887782358820700963961461289, 2.67878774145804997166649562950, 3.13772720771022185391582059998, 3.47257796576051962867658091826, 3.64586933829706876456116285702, 4.36113260080479830342227147056, 4.51227245967404926376717183911, 5.14767662828047935785018067411, 5.22018031788748319305832197204, 5.64424219206451047630738744027, 5.76893529579824842087721529913, 6.29731244939544276947778215530, 6.35749665209084826093271008217, 7.12412960075898860988496092559, 7.20631049385956534606659243840, 7.65489168011193579071417810725, 7.66263692450090686510326776552

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