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Properties

Label 4-10e4-1.1-c1e2-0-2
Degree $4$
Conductor $10000$
Sign $1$
Analytic cond. $0.637608$
Root an. cond. $0.893590$
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  + 2·9-s + 8·19-s − 12·29-s − 8·31-s + 12·41-s + 10·49-s − 24·59-s + 4·61-s − 24·71-s − 16·79-s − 5·81-s + 12·89-s + 12·101-s − 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 16·171-s + 173-s + ⋯
L(s)  = 1  + 2/3·9-s + 1.83·19-s − 2.22·29-s − 1.43·31-s + 1.87·41-s + 10/7·49-s − 3.12·59-s + 0.512·61-s − 2.84·71-s − 1.80·79-s − 5/9·81-s + 1.27·89-s + 1.19·101-s − 0.383·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 1.22·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10000\)    =    \(2^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.637608\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.034365470\)
\(L(\frac12)\) \(\approx\) \(1.034365470\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.47.a_acg
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.59.y_kc
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.73.a_afm
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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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

−14.33024189415308501730553576776, −13.52643273303449432817387322879, −12.98781540167559015103629201578, −12.81002754851319816436699890857, −11.81746361245409757385621519176, −11.75083487116390041285185985182, −10.78630433012170936477774500537, −10.62485914808997747602449648235, −9.678494221603797481097103622902, −9.284859191225383023872534913133, −8.960199626410135445348724291747, −7.79001924966433881829146023073, −7.43493432744747467519834720266, −7.11279555676085094514050237941, −5.81551045176731723788667961676, −5.68482641200058823791632445172, −4.61840726780746961176124773780, −3.87154079940685422623428227102, −3.00312250730576770700466962261, −1.62617164730213689031770455141, 1.62617164730213689031770455141, 3.00312250730576770700466962261, 3.87154079940685422623428227102, 4.61840726780746961176124773780, 5.68482641200058823791632445172, 5.81551045176731723788667961676, 7.11279555676085094514050237941, 7.43493432744747467519834720266, 7.79001924966433881829146023073, 8.960199626410135445348724291747, 9.284859191225383023872534913133, 9.678494221603797481097103622902, 10.62485914808997747602449648235, 10.78630433012170936477774500537, 11.75083487116390041285185985182, 11.81746361245409757385621519176, 12.81002754851319816436699890857, 12.98781540167559015103629201578, 13.52643273303449432817387322879, 14.33024189415308501730553576776

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