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Properties

Label 4-1050e2-1.1-c1e2-0-9
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 − 10·13-s − 14-s − 16-s − 6·17-s + 7·19-s + 21-s − 6·23-s + 24-s + 10·26-s − 27-s − 8·31-s + 6·34-s − 37-s − 7·38-s − 10·39-s − 42-s − 16·43-s + 6·46-s + 6·47-s − 48-s − 6·49-s − 6·51-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2.77·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s + 1.60·19-s + 0.218·21-s − 1.25·23-s + 0.204·24-s + 1.96·26-s − 0.192·27-s − 1.43·31-s + 1.02·34-s − 0.164·37-s − 1.13·38-s − 1.60·39-s − 0.154·42-s − 2.43·43-s + 0.884·46-s + 0.875·47-s − 0.144·48-s − 6/7·49-s − 0.840·51-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

\(L(1)\) \(\approx\) \(0.6355574783\)
\(L(\frac12)\) \(\approx\) \(0.6355574783\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.13.k_bz
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_t
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ah_be
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_n
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_bh
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.b_abk
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
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.ag_al
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$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.ab_aci
67$C_2^2$ \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.67.n_dy
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 - 5 T - 48 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.73.af_abw
79$C_2^2$ \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.79.ah_abe
83$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \) 2.83.abk_sw
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_acb
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

−10.02703468974041877185251715036, −9.525766818303533896474423004274, −9.239803040726290765750313419768, −9.204933924759514965685025924644, −8.405253032360885644011994647749, −7.88753918827989874547607118196, −7.87149881954552970509503110321, −7.38364876412076295594692138021, −6.81607552527801770307819077713, −6.67215301209161030783520210664, −5.82521485263371149903106564716, −5.18095264175520696050491607986, −4.79711603892737432456889712207, −4.77488490657734986152979262426, −3.59435046012128353016069701723, −3.57009651550726298748689200091, −2.45169380120769233031150109519, −2.26253071491922606438835554724, −1.67875849839359839785987381729, −0.37162367235802753064765556360, 0.37162367235802753064765556360, 1.67875849839359839785987381729, 2.26253071491922606438835554724, 2.45169380120769233031150109519, 3.57009651550726298748689200091, 3.59435046012128353016069701723, 4.77488490657734986152979262426, 4.79711603892737432456889712207, 5.18095264175520696050491607986, 5.82521485263371149903106564716, 6.67215301209161030783520210664, 6.81607552527801770307819077713, 7.38364876412076295594692138021, 7.87149881954552970509503110321, 7.88753918827989874547607118196, 8.405253032360885644011994647749, 9.204933924759514965685025924644, 9.239803040726290765750313419768, 9.525766818303533896474423004274, 10.02703468974041877185251715036

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