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Properties

Label 4-1050e2-1.1-c1e2-0-20
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 6-s + 7-s + 8-s + 8·13-s − 14-s − 16-s + 3·17-s − 2·19-s + 21-s + 3·23-s + 24-s − 8·26-s − 27-s + 31-s − 3·34-s − 10·37-s + 2·38-s + 8·39-s − 18·41-s − 42-s + 20·43-s − 3·46-s − 3·47-s − 48-s − 6·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 2.21·13-s − 0.267·14-s − 1/4·16-s + 0.727·17-s − 0.458·19-s + 0.218·21-s + 0.625·23-s + 0.204·24-s − 1.56·26-s − 0.192·27-s + 0.179·31-s − 0.514·34-s − 1.64·37-s + 0.324·38-s + 1.28·39-s − 2.81·41-s − 0.154·42-s + 3.04·43-s − 0.442·46-s − 0.437·47-s − 0.144·48-s − 6/7·49-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

\(L(1)\) \(\approx\) \(1.940690661\)
\(L(\frac12)\) \(\approx\) \(1.940690661\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
good11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ap
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_ao
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) 2.31.ab_abe
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.k_cl
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.41.s_gh
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.43.au_he
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.47.d_abm
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ar
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_ax
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_d
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_abz
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.71.ag_fv
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.73.ao_et
79$C_2^2$ \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.79.l_bq
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.89.ap_fg
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.97.ao_jj
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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

−10.10947634376912323818998725279, −9.537207033385270312146700296465, −9.194788683131538717220925477104, −8.865427652376597717908154681584, −8.467758320623969617667404430442, −8.127967439321144871985449707932, −7.951339805013917945808944643196, −7.30719702755385735801769452971, −6.78803349871044685627675805169, −6.43220105404141490793815228352, −5.96979657446936058243490264008, −5.22342493064881479518599758423, −5.16708892597076743679681653307, −4.14743461463550331514805346172, −3.97401909643869250000007308463, −3.21278390591509335972729746712, −3.01674915501339647834972413811, −1.84949279886908347444584635255, −1.58704811676151821398371958723, −0.73073652076445170815064283498, 0.73073652076445170815064283498, 1.58704811676151821398371958723, 1.84949279886908347444584635255, 3.01674915501339647834972413811, 3.21278390591509335972729746712, 3.97401909643869250000007308463, 4.14743461463550331514805346172, 5.16708892597076743679681653307, 5.22342493064881479518599758423, 5.96979657446936058243490264008, 6.43220105404141490793815228352, 6.78803349871044685627675805169, 7.30719702755385735801769452971, 7.951339805013917945808944643196, 8.127967439321144871985449707932, 8.467758320623969617667404430442, 8.865427652376597717908154681584, 9.194788683131538717220925477104, 9.537207033385270312146700296465, 10.10947634376912323818998725279

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