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Properties

Label 4-630e2-1.1-c1e2-0-41
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 + 4·7-s + 16-s + 3·17-s − 3·20-s + 4·25-s − 4·28-s + 12·35-s − 4·37-s + 3·41-s + 5·43-s + 3·47-s + 9·49-s − 9·59-s − 64-s + 16·67-s − 3·68-s − 5·79-s + 3·80-s + 3·83-s + 9·85-s − 3·89-s − 4·100-s − 15·101-s + 2·109-s + 4·112-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.34·5-s + 1.51·7-s + 1/4·16-s + 0.727·17-s − 0.670·20-s + 4/5·25-s − 0.755·28-s + 2.02·35-s − 0.657·37-s + 0.468·41-s + 0.762·43-s + 0.437·47-s + 9/7·49-s − 1.17·59-s − 1/8·64-s + 1.95·67-s − 0.363·68-s − 0.562·79-s + 0.335·80-s + 0.329·83-s + 0.976·85-s − 0.317·89-s − 2/5·100-s − 1.49·101-s + 0.191·109-s + 0.377·112-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

\(L(1)\) \(\approx\) \(2.773159893\)
\(L(\frac12)\) \(\approx\) \(2.773159893\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 - 3 T + p T^{2} \)
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$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \) 2.13.a_al
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ad_bi
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.a_al
23$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.23.a_ae
29$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.29.a_c
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.e_ad
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.41.ad_cm
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.43.af_dm
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.ad_ao
53$C_2^2$ \( 1 - 52 T^{2} + p^{2} T^{4} \) 2.53.a_aca
59$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.j_fg
61$C_2^2$ \( 1 - 89 T^{2} + p^{2} T^{4} \) 2.61.a_adl
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.67.aq_gg
71$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.71.a_ae
73$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \) 2.73.a_br
79$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.f_gg
83$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.83.ad_gk
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.89.d_ac
97$C_2^2$ \( 1 - 113 T^{2} + p^{2} T^{4} \) 2.97.a_aej
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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

−8.700499801780849871422967939717, −8.185872433937449181373744212163, −7.87526718901971601628891339895, −7.32216384434902412784033120957, −6.84588299918946371685968876469, −6.13339703928880059857534572843, −5.75739321044061657521912351317, −5.30466503180839410864620008210, −4.96691491474515561658927085928, −4.37697657818427220392102837937, −3.83342422335483417429324076111, −3.01087461802906842246889200636, −2.28332519884053315036700805030, −1.70730787434561622793059029794, −1.02391041856488217697902128492, 1.02391041856488217697902128492, 1.70730787434561622793059029794, 2.28332519884053315036700805030, 3.01087461802906842246889200636, 3.83342422335483417429324076111, 4.37697657818427220392102837937, 4.96691491474515561658927085928, 5.30466503180839410864620008210, 5.75739321044061657521912351317, 6.13339703928880059857534572843, 6.84588299918946371685968876469, 7.32216384434902412784033120957, 7.87526718901971601628891339895, 8.185872433937449181373744212163, 8.700499801780849871422967939717

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