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Properties

Label 4-5227200-1.1-c1e2-0-1
Degree $4$
Conductor $5227200$
Sign $1$
Analytic cond. $333.290$
Root an. cond. $4.27273$
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  − 3-s − 5-s + 9-s − 6·11-s + 15-s − 4·23-s − 4·25-s − 27-s + 13·31-s + 6·33-s − 2·37-s − 45-s − 4·47-s − 6·49-s − 5·53-s + 6·55-s + 10·59-s − 23·67-s + 4·69-s − 17·71-s + 4·75-s + 81-s − 7·89-s − 13·93-s − 6·99-s − 103-s + 2·111-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.80·11-s + 0.258·15-s − 0.834·23-s − 4/5·25-s − 0.192·27-s + 2.33·31-s + 1.04·33-s − 0.328·37-s − 0.149·45-s − 0.583·47-s − 6/7·49-s − 0.686·53-s + 0.809·55-s + 1.30·59-s − 2.80·67-s + 0.481·69-s − 2.01·71-s + 0.461·75-s + 1/9·81-s − 0.741·89-s − 1.34·93-s − 0.603·99-s − 0.0985·103-s + 0.189·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5227200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5227200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5227200\)    =    \(2^{6} \cdot 3^{3} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(333.290\)
Root analytic conductor: \(4.27273\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5227200,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3439485797\)
\(L(\frac12)\) \(\approx\) \(0.3439485797\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( 1 + T \)
5$C_2$ \( 1 + T + p T^{2} \)
11$C_2$ \( 1 + 6 T + p T^{2} \)
good7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.13.a_ab
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.17.a_q
19$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \) 2.19.a_y
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.e_bu
29$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.29.a_be
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.31.an_do
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.c_co
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.41.a_ck
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.47.e_cv
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.f_cs
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.59.ak_fm
61$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \) 2.61.a_aeb
67$C_2$$\times$$C_2$ \( ( 1 + 11 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.x_kg
71$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.71.r_ig
73$C_2^2$ \( 1 + 125 T^{2} + p^{2} T^{4} \) 2.73.a_ev
79$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.79.a_abi
83$C_2^2$ \( 1 - 103 T^{2} + p^{2} T^{4} \) 2.83.a_adz
89$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.89.h_cg
97$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.97.a_ack
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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

−7.32810792519462363579075763410, −6.97037650469573923996379116896, −6.34843206146956534720107199975, −6.08391177184320278685537348690, −5.68972314928779128367651658754, −5.30111686401588385575084776141, −4.68881636389280414427499859108, −4.54078731609562846440637049070, −4.11969489158323157066783943173, −3.30650229130426370682651886817, −3.05161483995839953138273129426, −2.47327378862423664395573636931, −1.91173304452267456152391344527, −1.19049732986691411995992225935, −0.22099173962176644626687841554, 0.22099173962176644626687841554, 1.19049732986691411995992225935, 1.91173304452267456152391344527, 2.47327378862423664395573636931, 3.05161483995839953138273129426, 3.30650229130426370682651886817, 4.11969489158323157066783943173, 4.54078731609562846440637049070, 4.68881636389280414427499859108, 5.30111686401588385575084776141, 5.68972314928779128367651658754, 6.08391177184320278685537348690, 6.34843206146956534720107199975, 6.97037650469573923996379116896, 7.32810792519462363579075763410

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