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Properties

Label 4-630e2-1.1-c1e2-0-51
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

\(L(1)\) \(\approx\) \(3.319674640\)
\(L(\frac12)\) \(\approx\) \(3.319674640\)
\(L(\frac{3}{2})\) 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$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ae_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$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ae_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$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.i_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$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.q_ig
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.m_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

−8.557062975195138528339974855083, −8.207114208582584696728360323356, −7.73718983337907671886956340301, −7.37026739180275891403332622063, −6.75439842094278852773351280866, −6.36881766379346555400116849040, −5.71892785760104327118877728605, −5.32566682982525059627463950650, −5.02383395829724229852490627586, −4.34739903650599522470128861177, −3.70577795729173186507963442792, −2.97749107347859756807500302237, −2.31735159116341016090358415673, −1.70835897226906595791475184002, −1.13607679413910264694535329196, 1.13607679413910264694535329196, 1.70835897226906595791475184002, 2.31735159116341016090358415673, 2.97749107347859756807500302237, 3.70577795729173186507963442792, 4.34739903650599522470128861177, 5.02383395829724229852490627586, 5.32566682982525059627463950650, 5.71892785760104327118877728605, 6.36881766379346555400116849040, 6.75439842094278852773351280866, 7.37026739180275891403332622063, 7.73718983337907671886956340301, 8.207114208582584696728360323356, 8.557062975195138528339974855083

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