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Properties

Label 4-650e2-1.1-c1e2-0-7
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4-s − 3·9-s − 2·12-s − 4·13-s + 16-s − 6·17-s + 12·23-s − 14·27-s + 3·36-s − 8·39-s + 2·43-s + 2·48-s + 5·49-s − 12·51-s + 4·52-s + 12·53-s − 16·61-s − 64-s + 6·68-s + 24·69-s + 20·79-s − 4·81-s − 12·92-s + 24·101-s − 28·103-s + 24·107-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s − 9-s − 0.577·12-s − 1.10·13-s + 1/4·16-s − 1.45·17-s + 2.50·23-s − 2.69·27-s + 1/2·36-s − 1.28·39-s + 0.304·43-s + 0.288·48-s + 5/7·49-s − 1.68·51-s + 0.554·52-s + 1.64·53-s − 2.04·61-s − 1/8·64-s + 0.727·68-s + 2.88·69-s + 2.25·79-s − 4/9·81-s − 1.25·92-s + 2.38·101-s − 2.75·103-s + 2.32·107-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

\(L(1)\) \(\approx\) \(1.584030363\)
\(L(\frac12)\) \(\approx\) \(1.584030363\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.3.ac_h
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
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$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.37.a_acn
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.43.ac_dj
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.47.a_adh
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
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$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.67.a_k
71$C_2^2$ \( 1 + 83 T^{2} + p^{2} T^{4} \) 2.71.a_df
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.89.a_afm
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.97.a_aby
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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

−10.89329599547418811356503150106, −10.34971898797595989320252739846, −9.605776554049539960768405289056, −9.308198244815035831406713849316, −9.002007262895981924125233786835, −8.682234634856913125841328137352, −8.407206674963692957500719941792, −7.64929583800892230370343949942, −7.47651389342887512601851415314, −6.85785597870050714067986650135, −6.39822660865749472858892430044, −5.60108455906016923837017513258, −5.37673738251340324102294189919, −4.59344199134021636518576752244, −4.41261918207217047788383709451, −3.29042029706112970569274377792, −3.24778115972941505072309752789, −2.43288778495280912154802728314, −2.10380209540382038011584865571, −0.61761418858325786133549348357, 0.61761418858325786133549348357, 2.10380209540382038011584865571, 2.43288778495280912154802728314, 3.24778115972941505072309752789, 3.29042029706112970569274377792, 4.41261918207217047788383709451, 4.59344199134021636518576752244, 5.37673738251340324102294189919, 5.60108455906016923837017513258, 6.39822660865749472858892430044, 6.85785597870050714067986650135, 7.47651389342887512601851415314, 7.64929583800892230370343949942, 8.407206674963692957500719941792, 8.682234634856913125841328137352, 9.002007262895981924125233786835, 9.308198244815035831406713849316, 9.605776554049539960768405289056, 10.34971898797595989320252739846, 10.89329599547418811356503150106

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