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Properties

Label 4-2250e2-1.1-c1e2-0-9
Degree $4$
Conductor $5062500$
Sign $1$
Analytic cond. $322.789$
Root an. cond. $4.23867$
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 + 3·4-s − 2·7-s − 4·8-s − 5·11-s + 13-s + 4·14-s + 5·16-s − 3·17-s + 12·19-s + 10·22-s − 5·23-s − 2·26-s − 6·28-s − 3·29-s + 11·31-s − 6·32-s + 6·34-s − 9·37-s − 24·38-s + 2·41-s − 7·43-s − 15·44-s + 10·46-s − 15·47-s − 6·49-s + 3·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.755·7-s − 1.41·8-s − 1.50·11-s + 0.277·13-s + 1.06·14-s + 5/4·16-s − 0.727·17-s + 2.75·19-s + 2.13·22-s − 1.04·23-s − 0.392·26-s − 1.13·28-s − 0.557·29-s + 1.97·31-s − 1.06·32-s + 1.02·34-s − 1.47·37-s − 3.89·38-s + 0.312·41-s − 1.06·43-s − 2.26·44-s + 1.47·46-s − 2.18·47-s − 6/7·49-s + 0.416·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5062500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5062500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5062500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{6}\)
Sign: $1$
Analytic conductor: \(322.789\)
Root analytic conductor: \(4.23867\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5062500,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
good7$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_k
11$D_{4}$ \( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.11.f_bb
13$D_{4}$ \( 1 - T + 15 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_p
17$D_{4}$ \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_z
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.19.am_cw
23$D_{4}$ \( 1 + 5 T + 21 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.23.f_v
29$C_4$ \( 1 + 3 T - T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_ab
31$C_4$ \( 1 - 11 T + 81 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.31.al_dd
37$D_{4}$ \( 1 + 9 T + 83 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.37.j_df
41$D_{4}$ \( 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.41.ac_da
43$D_{4}$ \( 1 + 7 T + 87 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.43.h_dj
47$D_{4}$ \( 1 + 15 T + 139 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.47.p_fj
53$D_{4}$ \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.53.c_dy
59$D_{4}$ \( 1 + 15 T + 173 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.59.p_gr
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$D_{4}$ \( 1 + 7 T + 145 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.67.h_fp
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.71.a_ck
73$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.73.y_le
79$D_{4}$ \( 1 + 9 T + 167 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.79.j_gl
83$D_{4}$ \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_by
89$D_{4}$ \( 1 - 14 T + 222 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.89.ao_io
97$D_{4}$ \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.97.o_eo
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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

−8.691608414023366668377700633747, −8.562511298373062113772253461174, −8.004763655917466761197380850061, −7.84404525766828686298857248045, −7.28869948039457584355056727023, −7.22246131109067090656515075055, −6.53502378006631873031052257613, −6.23123350452961415445727793834, −5.85990314156372845464536633258, −5.38637278567257045536173418573, −4.82541711187656626012140723419, −4.62563029385015336959601123631, −3.53857303109943604517961477781, −3.32300054254378282924255183354, −2.85363740435088396620677432284, −2.51644164625968517510565982997, −1.53300057577466734582820406806, −1.37074453755131399877935977374, 0, 0, 1.37074453755131399877935977374, 1.53300057577466734582820406806, 2.51644164625968517510565982997, 2.85363740435088396620677432284, 3.32300054254378282924255183354, 3.53857303109943604517961477781, 4.62563029385015336959601123631, 4.82541711187656626012140723419, 5.38637278567257045536173418573, 5.85990314156372845464536633258, 6.23123350452961415445727793834, 6.53502378006631873031052257613, 7.22246131109067090656515075055, 7.28869948039457584355056727023, 7.84404525766828686298857248045, 8.004763655917466761197380850061, 8.562511298373062113772253461174, 8.691608414023366668377700633747

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