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Properties

Label 4-4600e2-1.1-c1e2-0-10
Degree $4$
Conductor $21160000$
Sign $1$
Analytic cond. $1349.17$
Root an. cond. $6.06062$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·7-s − 9-s − 8·11-s − 3·13-s − 2·17-s + 8·19-s + 2·21-s − 2·23-s − 13·29-s + 7·31-s + 8·33-s + 6·37-s + 3·39-s − 3·41-s + 10·43-s − 5·47-s + 6·49-s + 2·51-s − 8·53-s − 8·57-s + 4·59-s + 14·61-s + 2·63-s + 4·67-s + 2·69-s − 5·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s − 1/3·9-s − 2.41·11-s − 0.832·13-s − 0.485·17-s + 1.83·19-s + 0.436·21-s − 0.417·23-s − 2.41·29-s + 1.25·31-s + 1.39·33-s + 0.986·37-s + 0.480·39-s − 0.468·41-s + 1.52·43-s − 0.729·47-s + 6/7·49-s + 0.280·51-s − 1.09·53-s − 1.05·57-s + 0.520·59-s + 1.79·61-s + 0.251·63-s + 0.488·67-s + 0.240·69-s − 0.593·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21160000\)    =    \(2^{6} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1349.17\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 21160000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
23$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_c
7$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ac
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
13$D_{4}$ \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.13.d_y
17$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_s
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
29$D_{4}$ \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.29.n_ds
31$D_{4}$ \( 1 - 7 T + 70 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.31.ah_cs
37$D_{4}$ \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_co
41$D_{4}$ \( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_dc
43$D_{4}$ \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.43.ak_dq
47$D_{4}$ \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.47.f_ck
53$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_cc
59$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_cc
61$D_{4}$ \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.61.ao_fy
67$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_cs
71$D_{4}$ \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.71.f_eg
73$D_{4}$ \( 1 + 29 T + 352 T^{2} + 29 p T^{3} + p^{2} T^{4} \) 2.73.bd_no
79$D_{4}$ \( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.79.e_dq
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.97.aq_hi
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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

−8.043263029068261767989856582225, −7.52566892838366889086445582171, −7.50163501992219810866232658636, −7.29287365126195095101783479397, −6.68985401511206780447490330070, −6.19122537890131129570406385135, −5.83575618830576951362275515074, −5.49260115746555568794419120038, −5.34883898806530265468696558106, −5.01160260568204496222600984846, −4.26072686023711780495047812552, −4.25250360972244496520702328841, −3.34879110571241952300008360331, −3.10897367761189709719639502997, −2.57032539003755719809084489101, −2.45736458521699049868109411435, −1.71510230203107351462532255140, −0.891705102404798781509687668570, 0, 0, 0.891705102404798781509687668570, 1.71510230203107351462532255140, 2.45736458521699049868109411435, 2.57032539003755719809084489101, 3.10897367761189709719639502997, 3.34879110571241952300008360331, 4.25250360972244496520702328841, 4.26072686023711780495047812552, 5.01160260568204496222600984846, 5.34883898806530265468696558106, 5.49260115746555568794419120038, 5.83575618830576951362275515074, 6.19122537890131129570406385135, 6.68985401511206780447490330070, 7.29287365126195095101783479397, 7.50163501992219810866232658636, 7.52566892838366889086445582171, 8.043263029068261767989856582225

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