↑

Properties

Label 4-1050e2-1.1-c1e2-0-29
Degree $4$
Conductor $1102500$
Sign $1$
Analytic cond. $70.2963$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 6-s + 5·7-s − 8-s + 4·11-s − 2·13-s + 5·14-s − 16-s + 2·17-s − 19-s + 5·21-s + 4·22-s − 2·23-s − 24-s − 2·26-s − 27-s + 8·29-s + 4·33-s + 2·34-s + 3·37-s − 38-s − 2·39-s + 24·41-s + 5·42-s − 16·43-s − 2·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.408·6-s + 1.88·7-s − 0.353·8-s + 1.20·11-s − 0.554·13-s + 1.33·14-s − 1/4·16-s + 0.485·17-s − 0.229·19-s + 1.09·21-s + 0.852·22-s − 0.417·23-s − 0.204·24-s − 0.392·26-s − 0.192·27-s + 1.48·29-s + 0.696·33-s + 0.342·34-s + 0.493·37-s − 0.162·38-s − 0.320·39-s + 3.74·41-s + 0.771·42-s − 2.43·43-s − 0.294·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1102500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(70.2963\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1102500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.205501876\)
\(L(\frac12)\) \(\approx\) \(5.205501876\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_f
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.13.c_bb
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_an
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.b_as
23$C_2^2$ \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_at
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.29.ai_cw
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.31.a_abf
37$C_2^2$ \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.37.ad_abc
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.41.ay_is
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_al
53$C_2^2$ \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.53.ac_abx
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_ax
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.an_ee
67$C_2^2$ \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.67.ad_acg
71$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.71.abg_pi
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.73.l_bw
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.79.n_dm
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$C_2^2$ \( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.89.ac_adh
97$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \) 2.97.bi_sp
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.885402541530216065003883806669, −9.710295657921391609462350667901, −9.386260591263690445229289367104, −8.669389793481505132116177440196, −8.393791829652997053060695584901, −8.214843272729450627964479755401, −7.63393912654400044330095811871, −7.34575704552509271252165784771, −6.57803037360237864549288269393, −6.42786527467189148302351661791, −5.67426197382946463403343697812, −5.32332145085047594119610255988, −4.70736626634911097987615049385, −4.57209107021365080854470910218, −3.82154022124820655775315412367, −3.73877961714802317696799223125, −2.55557417139178764772792395553, −2.51914419268996084391199297535, −1.55159543623333042154548385138, −1.00012898623254346807919900663, 1.00012898623254346807919900663, 1.55159543623333042154548385138, 2.51914419268996084391199297535, 2.55557417139178764772792395553, 3.73877961714802317696799223125, 3.82154022124820655775315412367, 4.57209107021365080854470910218, 4.70736626634911097987615049385, 5.32332145085047594119610255988, 5.67426197382946463403343697812, 6.42786527467189148302351661791, 6.57803037360237864549288269393, 7.34575704552509271252165784771, 7.63393912654400044330095811871, 8.214843272729450627964479755401, 8.393791829652997053060695584901, 8.669389793481505132116177440196, 9.386260591263690445229289367104, 9.710295657921391609462350667901, 9.885402541530216065003883806669

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