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Properties

Label 4-837e2-1.1-c1e2-0-4
Degree $4$
Conductor $700569$
Sign $1$
Analytic cond. $44.6688$
Root an. cond. $2.58524$
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  − 4·4-s − 2·7-s − 2·13-s + 12·16-s + 10·19-s + 2·25-s + 8·28-s + 2·31-s + 10·37-s + 16·43-s − 11·49-s + 8·52-s + 10·61-s − 32·64-s + 10·67-s + 22·73-s − 40·76-s − 2·79-s + 4·91-s + 10·97-s − 8·100-s − 2·103-s − 20·109-s − 24·112-s − 10·121-s − 8·124-s + 127-s + ⋯
L(s)  = 1  − 2·4-s − 0.755·7-s − 0.554·13-s + 3·16-s + 2.29·19-s + 2/5·25-s + 1.51·28-s + 0.359·31-s + 1.64·37-s + 2.43·43-s − 1.57·49-s + 1.10·52-s + 1.28·61-s − 4·64-s + 1.22·67-s + 2.57·73-s − 4.58·76-s − 0.225·79-s + 0.419·91-s + 1.01·97-s − 4/5·100-s − 0.197·103-s − 1.91·109-s − 2.26·112-s − 0.909·121-s − 0.718·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(700569\)    =    \(3^{6} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(44.6688\)
Root analytic conductor: \(2.58524\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 700569,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.055631988\)
\(L(\frac12)\) \(\approx\) \(1.055631988\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
31$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.2.a_e
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.7.c_p
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.13.c_bb
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.17.a_w
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.19.ak_cl
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.37.ak_dv
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.41.a_cs
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.47.a_ao
53$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \) 2.53.a_dq
59$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.59.a_acw
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.61.ak_fr
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.67.ak_gd
71$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \) 2.71.a_dq
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.73.aw_kh
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.79.c_gd
83$C_2^2$ \( 1 + 154 T^{2} + p^{2} T^{4} \) 2.83.a_fy
89$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.89.a_cs
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.97.ak_il
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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

−10.03292564047177048993688046725, −9.911809846592584450323299545132, −9.463259026207943941620278475260, −9.277917035403339643756163323383, −9.030656670531920628561741815314, −8.153523071424416519971803161812, −7.988695506949047678660919363573, −7.67444565112721543409497728603, −7.01491595581626317194093531625, −6.51178611587823318733120043472, −5.82996264208230393975028440681, −5.53415097725298244112775365184, −4.93460111281073043164564862460, −4.79618401994606835315143031039, −3.89853889980200916525998639103, −3.78038830838795663436356938691, −3.03372768606895926394732302672, −2.56505452685828550606153552642, −1.14630803571439896169326558963, −0.63929328049410835138045782888, 0.63929328049410835138045782888, 1.14630803571439896169326558963, 2.56505452685828550606153552642, 3.03372768606895926394732302672, 3.78038830838795663436356938691, 3.89853889980200916525998639103, 4.79618401994606835315143031039, 4.93460111281073043164564862460, 5.53415097725298244112775365184, 5.82996264208230393975028440681, 6.51178611587823318733120043472, 7.01491595581626317194093531625, 7.67444565112721543409497728603, 7.988695506949047678660919363573, 8.153523071424416519971803161812, 9.030656670531920628561741815314, 9.277917035403339643756163323383, 9.463259026207943941620278475260, 9.911809846592584450323299545132, 10.03292564047177048993688046725

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