Properties

Label 4-1150e2-1.1-c1e2-0-8
Degree $4$
Conductor $1322500$
Sign $1$
Analytic cond. $84.3237$
Root an. cond. $3.03031$
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 − 3-s + 3·4-s + 2·6-s − 7-s − 4·8-s − 4·9-s + 11-s − 3·12-s + 3·13-s + 2·14-s + 5·16-s − 17-s + 8·18-s − 3·19-s + 21-s − 2·22-s − 2·23-s + 4·24-s − 6·26-s + 6·27-s − 3·28-s − 14·29-s + 7·31-s − 6·32-s − 33-s + 2·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 3/2·4-s + 0.816·6-s − 0.377·7-s − 1.41·8-s − 4/3·9-s + 0.301·11-s − 0.866·12-s + 0.832·13-s + 0.534·14-s + 5/4·16-s − 0.242·17-s + 1.88·18-s − 0.688·19-s + 0.218·21-s − 0.426·22-s − 0.417·23-s + 0.816·24-s − 1.17·26-s + 1.15·27-s − 0.566·28-s − 2.59·29-s + 1.25·31-s − 1.06·32-s − 0.174·33-s + 0.342·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1322500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1322500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1322500\)    =    \(2^{2} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(84.3237\)
Root analytic conductor: \(3.03031\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1322500,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
23$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_f
7$D_{4}$ \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \) 2.7.b_n
11$D_{4}$ \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_l
13$D_{4}$ \( 1 - 3 T - 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.13.ad_ad
17$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) 2.17.b_d
19$D_{4}$ \( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.19.d_bd
29$D_{4}$ \( 1 + 14 T + 102 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.29.o_dy
31$D_{4}$ \( 1 - 7 T + 43 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.31.ah_br
37$D_{4}$ \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_cg
41$D_{4}$ \( 1 + 9 T + p T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_bp
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$D_{4}$ \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_cg
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_dy
59$D_{4}$ \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.59.k_du
61$D_{4}$ \( 1 + 3 T + 63 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.61.d_cl
67$D_{4}$ \( 1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.67.u_ig
71$D_{4}$ \( 1 - 3 T + 113 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.71.ad_ej
73$D_{4}$ \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_fm
79$D_{4}$ \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_gs
83$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_dm
89$C_4$ \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_hm
97$D_{4}$ \( 1 + 27 T + 375 T^{2} + 27 p T^{3} + p^{2} T^{4} \) 2.97.bb_ol
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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

−9.489413930475761147024393436630, −9.295733110368483409577233262984, −8.614865746819952084068378711126, −8.582959192436207916173518428837, −7.921086310117812988666562924894, −7.901987380887807389085272989017, −6.95892987365109291345399365340, −6.77459977918644625875056233663, −6.23394833996723476352050564115, −6.07816567610664796363613436873, −5.37357833558689789638200397169, −5.25058136594089464018124516345, −4.19813928089961116308847248245, −3.82147116423406732730973347220, −2.95640737939359410198793263623, −2.87744013085417543521607280372, −1.78059550635149144342141258246, −1.46896043750359807264804449626, 0, 0, 1.46896043750359807264804449626, 1.78059550635149144342141258246, 2.87744013085417543521607280372, 2.95640737939359410198793263623, 3.82147116423406732730973347220, 4.19813928089961116308847248245, 5.25058136594089464018124516345, 5.37357833558689789638200397169, 6.07816567610664796363613436873, 6.23394833996723476352050564115, 6.77459977918644625875056233663, 6.95892987365109291345399365340, 7.901987380887807389085272989017, 7.921086310117812988666562924894, 8.582959192436207916173518428837, 8.614865746819952084068378711126, 9.295733110368483409577233262984, 9.489413930475761147024393436630

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