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Properties

Label 4-7350e2-1.1-c1e2-0-6
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 $2$

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 − 6·17-s − 6·18-s − 8·22-s + 2·23-s + 8·24-s + 12·26-s − 4·27-s + 2·29-s + 2·31-s − 6·32-s − 8·33-s + 12·34-s + 9·36-s + 12·39-s + 6·41-s + 6·43-s + 12·44-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 − 1.45·17-s − 1.41·18-s − 1.70·22-s + 0.417·23-s + 1.63·24-s + 2.35·26-s − 0.769·27-s + 0.371·29-s + 0.359·31-s − 1.06·32-s − 1.39·33-s + 2.05·34-s + 3/2·36-s + 1.92·39-s + 0.937·41-s + 0.914·43-s + 1.80·44-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: \(2\)
Selberg data: \((4,\ 54022500,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
7 \( 1 \)
good11$D_{4}$ \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_y
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.13.g_bj
17$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_bj
19$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \) 2.19.a_bk
23$D_{4}$ \( 1 - 2 T + 45 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_bt
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.29.ac_ch
31$D_{4}$ \( 1 - 2 T + 61 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_cj
37$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \) 2.37.a_y
41$D_{4}$ \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_df
43$D_{4}$ \( 1 - 6 T + 93 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_dp
47$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_da
53$D_{4}$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.53.ao_et
59$D_{4}$ \( 1 + 6 T + 109 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_ef
61$D_{4}$ \( 1 - 6 T + 123 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.61.ag_et
67$D_{4}$ \( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_w
71$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_s
73$D_{4}$ \( 1 + 16 T + 160 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.73.q_ge
79$D_{4}$ \( 1 + 12 T + 144 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_fo
83$D_{4}$ \( 1 + 14 T + 165 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.83.o_gj
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.89.i_hm
97$D_{4}$ \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.97.am_hq
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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

−7.44738457936252861439043351225, −7.42050719436148590251891713035, −6.97524019143703877152728860628, −6.95899990358277577653819572118, −6.33317022405491944313625919723, −6.23919894503354940519719772265, −5.72387497126071200410068388500, −5.44164303381253084263947924515, −4.91661923398124263917147781700, −4.49297742994058968925979669151, −4.25174530870666938447738743425, −3.93719486980068151461627468688, −3.00237178770594034357857191264, −2.90447206019731083618982246724, −2.10711473110370834063780098661, −2.07910301303764931853925493238, −1.10310008880618584160343873059, −1.09989353415292719631247914622, 0, 0, 1.09989353415292719631247914622, 1.10310008880618584160343873059, 2.07910301303764931853925493238, 2.10711473110370834063780098661, 2.90447206019731083618982246724, 3.00237178770594034357857191264, 3.93719486980068151461627468688, 4.25174530870666938447738743425, 4.49297742994058968925979669151, 4.91661923398124263917147781700, 5.44164303381253084263947924515, 5.72387497126071200410068388500, 6.23919894503354940519719772265, 6.33317022405491944313625919723, 6.95899990358277577653819572118, 6.97524019143703877152728860628, 7.42050719436148590251891713035, 7.44738457936252861439043351225

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