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Properties

Label 4-1850e2-1.1-c1e2-0-21
Degree $4$
Conductor $3422500$
Sign $1$
Analytic cond. $218.221$
Root an. cond. $3.84347$
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 + 3·4-s + 4·7-s + 4·8-s + 8·13-s + 8·14-s + 5·16-s + 12·19-s − 8·23-s + 16·26-s + 12·28-s − 8·31-s + 6·32-s + 2·37-s + 24·38-s + 4·41-s + 8·43-s − 16·46-s + 4·47-s + 4·49-s + 24·52-s − 4·53-s + 16·56-s − 12·59-s + 24·61-s − 16·62-s + 7·64-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.51·7-s + 1.41·8-s + 2.21·13-s + 2.13·14-s + 5/4·16-s + 2.75·19-s − 1.66·23-s + 3.13·26-s + 2.26·28-s − 1.43·31-s + 1.06·32-s + 0.328·37-s + 3.89·38-s + 0.624·41-s + 1.21·43-s − 2.35·46-s + 0.583·47-s + 4/7·49-s + 3.32·52-s − 0.549·53-s + 2.13·56-s − 1.56·59-s + 3.07·61-s − 2.03·62-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3422500\)    =    \(2^{2} \cdot 5^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(218.221\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3422500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(10.30179650\)
\(L(\frac12)\) \(\approx\) \(10.30179650\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
7$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_m
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.17.a_k
19$D_{4}$ \( 1 - 12 T + 68 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.19.am_cq
23$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.23.i_bm
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$D_{4}$ \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_cu
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$D_{4}$ \( 1 - 4 T + 92 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_do
53$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.53.e_o
59$D_{4}$ \( 1 + 12 T + 148 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_fs
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.61.ay_kg
67$D_{4}$ \( 1 + 16 T + 192 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.67.q_hk
71$C_2^2$ \( 1 + 118 T^{2} + p^{2} T^{4} \) 2.71.a_eo
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.73.ai_gg
79$D_{4}$ \( 1 - 8 T + 168 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.79.ai_gm
83$D_{4}$ \( 1 + 24 T + 304 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.83.y_ls
89$D_{4}$ \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_eo
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_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

−9.240931339682008664823811233368, −9.235291918252385620770789855389, −8.351904345703161987307602902443, −8.266502118590967936091875153892, −7.72426871477836410663507883220, −7.48367395725392788651788867020, −7.14284119065304406861776878814, −6.45698737624112050522683710410, −6.03523087644994316540443736665, −5.74665205150198984708597417267, −5.29676547057431203415337273547, −5.19176673177871348612719735805, −4.40385530484308334981962436257, −3.96845553263633615807602578003, −3.84452367022871124691960758388, −3.20692281625611235139427726240, −2.72530765844196990624579879335, −1.97516987234776102347959159132, −1.40293693537768900664995975095, −1.10859637255207741574631887087, 1.10859637255207741574631887087, 1.40293693537768900664995975095, 1.97516987234776102347959159132, 2.72530765844196990624579879335, 3.20692281625611235139427726240, 3.84452367022871124691960758388, 3.96845553263633615807602578003, 4.40385530484308334981962436257, 5.19176673177871348612719735805, 5.29676547057431203415337273547, 5.74665205150198984708597417267, 6.03523087644994316540443736665, 6.45698737624112050522683710410, 7.14284119065304406861776878814, 7.48367395725392788651788867020, 7.72426871477836410663507883220, 8.266502118590967936091875153892, 8.351904345703161987307602902443, 9.235291918252385620770789855389, 9.240931339682008664823811233368

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