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Properties

Label 4-630e2-1.1-c1e2-0-19
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 − 3·5-s + 2·7-s + 16-s − 3·17-s − 3·20-s + 4·25-s + 2·28-s − 6·35-s + 4·37-s − 9·41-s + 7·43-s + 3·47-s − 3·49-s − 9·59-s + 64-s − 2·67-s − 3·68-s + 13·79-s − 3·80-s + 9·83-s + 9·85-s + 27·89-s + 4·100-s + 21·101-s + 10·109-s + 2·112-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.34·5-s + 0.755·7-s + 1/4·16-s − 0.727·17-s − 0.670·20-s + 4/5·25-s + 0.377·28-s − 1.01·35-s + 0.657·37-s − 1.40·41-s + 1.06·43-s + 0.437·47-s − 3/7·49-s − 1.17·59-s + 1/8·64-s − 0.244·67-s − 0.363·68-s + 1.46·79-s − 0.335·80-s + 0.987·83-s + 0.976·85-s + 2.86·89-s + 2/5·100-s + 2.08·101-s + 0.957·109-s + 0.188·112-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

\(L(1)\) \(\approx\) \(1.516493768\)
\(L(\frac12)\) \(\approx\) \(1.516493768\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 3 T + p T^{2} \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
good11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.a_b
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.19.a_ax
23$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.23.a_abg
29$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.29.a_q
31$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.31.a_ao
37$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.37.ae_cr
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.j_de
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.43.ah_da
47$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.ad_dq
53$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \) 2.53.a_ado
59$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.j_fg
61$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.61.a_cd
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.c_ew
71$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.71.a_bo
73$C_2^2$ \( 1 + 115 T^{2} + p^{2} T^{4} \) 2.73.a_el
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.79.an_hq
83$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.aj_cy
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) 2.89.abb_nu
97$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.97.a_t
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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

−8.556493239835276697590993097578, −8.039922340386931372098788291605, −7.68794886483652809504912275802, −7.54251869422024537782489436757, −6.78163432702886401313762831784, −6.48630338069938182722751137406, −5.89752417964332046465351622330, −5.23378862434899764945376695841, −4.61393396824419085033327398425, −4.44300672339463317384217423615, −3.59860794598074088361512451480, −3.29724093437347876856199244629, −2.38783805629633132430519997483, −1.79511708048681920843006667623, −0.68196528440349150030179480817, 0.68196528440349150030179480817, 1.79511708048681920843006667623, 2.38783805629633132430519997483, 3.29724093437347876856199244629, 3.59860794598074088361512451480, 4.44300672339463317384217423615, 4.61393396824419085033327398425, 5.23378862434899764945376695841, 5.89752417964332046465351622330, 6.48630338069938182722751137406, 6.78163432702886401313762831784, 7.54251869422024537782489436757, 7.68794886483652809504912275802, 8.039922340386931372098788291605, 8.556493239835276697590993097578

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