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Properties

Label 4-2900e2-1.1-c1e2-0-7
Degree $4$
Conductor $8410000$
Sign $1$
Analytic cond. $536.228$
Root an. cond. $4.81213$
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  + 6·9-s − 4·11-s + 4·19-s − 2·29-s + 4·31-s − 20·41-s + 14·49-s + 24·59-s − 20·61-s + 24·71-s − 4·79-s + 27·81-s + 20·89-s − 24·99-s − 12·101-s − 12·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯
L(s)  = 1  + 2·9-s − 1.20·11-s + 0.917·19-s − 0.371·29-s + 0.718·31-s − 3.12·41-s + 2·49-s + 3.12·59-s − 2.56·61-s + 2.84·71-s − 0.450·79-s + 3·81-s + 2.11·89-s − 2.41·99-s − 1.19·101-s − 1.14·109-s − 0.909·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8410000\)    =    \(2^{4} \cdot 5^{4} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(536.228\)
Root analytic conductor: \(4.81213\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8410000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.904990388\)
\(L(\frac12)\) \(\approx\) \(2.904990388\)
\(L(\frac{3}{2})\) 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 \( 1 \)
29$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.3.a_ag
7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.17.a_abi
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.37.a_acg
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
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.ay_kc
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.67.a_k
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.73.a_ac
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.79.e_gg
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−9.052978364136340422138182232645, −8.453442722646973244918741045268, −8.000137497253049560981281616718, −7.981080650961969339112582302446, −7.40905058891638774434677971101, −7.02680600150139887846831327106, −6.77931181155221894837203349012, −6.54379005075471290013237555818, −5.76455817118344703199740287192, −5.41279940944279165696937535638, −5.04591397138533902146460590538, −4.80493460468917129426563953448, −4.15731252643609686278314349118, −3.88751859682147661393167948392, −3.36132623144193292438649194868, −2.92133659661270515027024579327, −2.18101251717164426773249743024, −1.90406125190320875997852166940, −1.18601639715493187244629271629, −0.59435553238787811240763273639, 0.59435553238787811240763273639, 1.18601639715493187244629271629, 1.90406125190320875997852166940, 2.18101251717164426773249743024, 2.92133659661270515027024579327, 3.36132623144193292438649194868, 3.88751859682147661393167948392, 4.15731252643609686278314349118, 4.80493460468917129426563953448, 5.04591397138533902146460590538, 5.41279940944279165696937535638, 5.76455817118344703199740287192, 6.54379005075471290013237555818, 6.77931181155221894837203349012, 7.02680600150139887846831327106, 7.40905058891638774434677971101, 7.981080650961969339112582302446, 8.000137497253049560981281616718, 8.453442722646973244918741045268, 9.052978364136340422138182232645

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