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Properties

Label 4-630e2-1.1-c1e2-0-25
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 − 4·17-s − 2·20-s + 3·25-s + 4·28-s − 8·35-s − 4·37-s − 4·41-s + 8·47-s + 9·49-s + 64-s + 16·67-s − 4·68-s + 8·79-s − 2·80-s + 16·83-s + 8·85-s + 12·89-s + 3·100-s − 4·101-s − 4·109-s + 4·112-s − 16·119-s + 10·121-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.894·5-s + 1.51·7-s + 1/4·16-s − 0.970·17-s − 0.447·20-s + 3/5·25-s + 0.755·28-s − 1.35·35-s − 0.657·37-s − 0.624·41-s + 1.16·47-s + 9/7·49-s + 1/8·64-s + 1.95·67-s − 0.485·68-s + 0.900·79-s − 0.223·80-s + 1.75·83-s + 0.867·85-s + 1.27·89-s + 3/10·100-s − 0.398·101-s − 0.383·109-s + 0.377·112-s − 1.46·119-s + 0.909·121-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

\(L(1)\) \(\approx\) \(2.015557201\)
\(L(\frac12)\) \(\approx\) \(2.015557201\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.e_bi
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.19.a_ba
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.31.a_be
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.e_bq
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.e_cs
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.ai_cw
53$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.53.a_ak
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.61.a_ag
67$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.67.aq_hm
71$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.71.a_ao
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.79.ai_gc
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.83.aq_ig
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.am_fu
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \) 2.97.a_afq
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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

−8.488666169660922217787232246613, −8.247959970208927584215423063391, −7.64965867242811473700883674168, −7.47612137655443434059671877263, −6.78692096022600538040428145862, −6.54221759243591394221297827494, −5.79146513302088817770129748984, −5.13633202884306868651164112443, −4.91171317203703943434316291614, −4.23588003054570288524627127566, −3.82256987971119378315316667466, −3.14550509008394024515126365733, −2.27176631580022225619805999882, −1.85640153540371798740730387983, −0.799981557337998082032325148331, 0.799981557337998082032325148331, 1.85640153540371798740730387983, 2.27176631580022225619805999882, 3.14550509008394024515126365733, 3.82256987971119378315316667466, 4.23588003054570288524627127566, 4.91171317203703943434316291614, 5.13633202884306868651164112443, 5.79146513302088817770129748984, 6.54221759243591394221297827494, 6.78692096022600538040428145862, 7.47612137655443434059671877263, 7.64965867242811473700883674168, 8.247959970208927584215423063391, 8.488666169660922217787232246613

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