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Properties

Label 4-630e2-1.1-c1e2-0-12
Degree $4$
Conductor $396900$
Sign $1$
Analytic cond. $25.3066$
Root an. cond. $2.24289$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 2·5-s − 4·7-s + 16-s + 8·17-s − 2·20-s − 25-s − 4·28-s + 8·35-s + 8·37-s − 8·41-s + 9·49-s − 20·59-s + 64-s + 8·68-s + 24·79-s − 2·80-s + 16·83-s − 16·85-s − 16·89-s − 100-s − 12·101-s − 20·109-s − 4·112-s − 32·119-s + 6·121-s + 12·125-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.894·5-s − 1.51·7-s + 1/4·16-s + 1.94·17-s − 0.447·20-s − 1/5·25-s − 0.755·28-s + 1.35·35-s + 1.31·37-s − 1.24·41-s + 9/7·49-s − 2.60·59-s + 1/8·64-s + 0.970·68-s + 2.70·79-s − 0.223·80-s + 1.75·83-s − 1.73·85-s − 1.69·89-s − 0.0999·100-s − 1.19·101-s − 1.91·109-s − 0.377·112-s − 2.93·119-s + 6/11·121-s + 1.07·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(396900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(25.3066\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 396900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.222625470\)
\(L(\frac12)\) \(\approx\) \(1.222625470\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.17.ai_by
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.19.a_o
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.23.a_abi
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.29.a_abm
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ai_cc
41$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.i_bi
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.a_da
53$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.53.a_acg
59$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.59.u_hu
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.61.a_ec
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.71.a_ek
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.73.a_s
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.79.ay_lq
83$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.83.aq_is
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.q_gw
97$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.97.a_de
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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

−8.526304573354618889859551744321, −8.042348491847199751941450893204, −7.74045876955467714798582623577, −7.36458787254134938292452182615, −6.75477726662937101297022639377, −6.40286336252377882114135323366, −5.90668267447385309393848192863, −5.47690816377993950303524300391, −4.80040668774363214792997040300, −4.10250727905773115014405068812, −3.53147883433701775898595880006, −3.21423536413491151318727725736, −2.71868394026060869490662576144, −1.67275795887950100619880491148, −0.61262454905339192166777130008, 0.61262454905339192166777130008, 1.67275795887950100619880491148, 2.71868394026060869490662576144, 3.21423536413491151318727725736, 3.53147883433701775898595880006, 4.10250727905773115014405068812, 4.80040668774363214792997040300, 5.47690816377993950303524300391, 5.90668267447385309393848192863, 6.40286336252377882114135323366, 6.75477726662937101297022639377, 7.36458787254134938292452182615, 7.74045876955467714798582623577, 8.042348491847199751941450893204, 8.526304573354618889859551744321

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