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Properties

Label 4-650e2-1.1-c1e2-0-31
Degree $4$
Conductor $422500$
Sign $1$
Analytic cond. $26.9389$
Root an. cond. $2.27821$
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  + 2·2-s + 3·4-s + 6·7-s + 4·8-s + 5·9-s − 6·13-s + 12·14-s + 5·16-s + 10·18-s − 12·26-s + 18·28-s + 6·32-s + 15·36-s + 6·37-s + 6·47-s + 13·49-s − 18·52-s + 24·56-s − 16·61-s + 30·63-s + 7·64-s − 24·67-s + 20·72-s − 12·73-s + 12·74-s − 20·79-s + 16·81-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 2.26·7-s + 1.41·8-s + 5/3·9-s − 1.66·13-s + 3.20·14-s + 5/4·16-s + 2.35·18-s − 2.35·26-s + 3.40·28-s + 1.06·32-s + 5/2·36-s + 0.986·37-s + 0.875·47-s + 13/7·49-s − 2.49·52-s + 3.20·56-s − 2.04·61-s + 3.77·63-s + 7/8·64-s − 2.93·67-s + 2.35·72-s − 1.40·73-s + 1.39·74-s − 2.25·79-s + 16/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(422500\)    =    \(2^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(26.9389\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 422500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.814025364\)
\(L(\frac12)\) \(\approx\) \(6.814025364\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.3.a_af
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.7.ag_x
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.17.a_az
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.37.ag_df
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.43.a_adh
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.47.ag_dz
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.59.a_ade
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2^2$ \( 1 + 83 T^{2} + p^{2} T^{4} \) 2.71.a_df
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.89.a_afm
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.97.y_na
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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

−10.77128747249240268522110685343, −10.59358246500303985706051099521, −9.919582178195877658614606755822, −9.792805258678974609615639773783, −8.971911149774787757747833746440, −8.497338597093787120376714571584, −7.77321445513339314779217604491, −7.61891432149947105942559238352, −7.18900165627792106120454499034, −7.00401701590232774144305270559, −5.93614338951729146501397390812, −5.83743036584630238830744322415, −4.87140494897237862716478860139, −4.81601720941620821500807048744, −4.36530358712088292472820094178, −4.16491894389083767687205950364, −3.07387344338991951809724368298, −2.51421172475490529448825642946, −1.65692696845448470095164131922, −1.47710430472257870417924395030, 1.47710430472257870417924395030, 1.65692696845448470095164131922, 2.51421172475490529448825642946, 3.07387344338991951809724368298, 4.16491894389083767687205950364, 4.36530358712088292472820094178, 4.81601720941620821500807048744, 4.87140494897237862716478860139, 5.83743036584630238830744322415, 5.93614338951729146501397390812, 7.00401701590232774144305270559, 7.18900165627792106120454499034, 7.61891432149947105942559238352, 7.77321445513339314779217604491, 8.497338597093787120376714571584, 8.971911149774787757747833746440, 9.792805258678974609615639773783, 9.919582178195877658614606755822, 10.59358246500303985706051099521, 10.77128747249240268522110685343

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