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Properties

Label 4-792e2-1.1-c1e2-0-12
Degree $4$
Conductor $627264$
Sign $1$
Analytic cond. $39.9948$
Root an. cond. $2.51478$
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  − 4·5-s + 4·11-s + 6·25-s + 8·31-s − 4·37-s + 16·47-s − 2·49-s − 4·53-s − 16·55-s + 8·59-s − 4·89-s − 4·97-s − 8·103-s − 4·113-s + 5·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 32·155-s + 157-s + 163-s + 167-s + 6·169-s + ⋯
L(s)  = 1  − 1.78·5-s + 1.20·11-s + 6/5·25-s + 1.43·31-s − 0.657·37-s + 2.33·47-s − 2/7·49-s − 0.549·53-s − 2.15·55-s + 1.04·59-s − 0.423·89-s − 0.406·97-s − 0.788·103-s − 0.376·113-s + 5/11·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 2.57·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6/13·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 627264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(627264\)    =    \(2^{6} \cdot 3^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(39.9948\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 627264,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.228178605\)
\(L(\frac12)\) \(\approx\) \(1.228178605\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.e_k
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.19.a_g
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.e_ck
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.41.a_aby
43$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.43.a_bm
47$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.47.aq_fy
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.53.e_cw
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.ai_eo
61$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.61.a_acs
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.a_fi
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.79.a_s
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.83.a_adm
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.e_acw
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−8.343180798202676655298184263414, −7.87524926693289186748273169556, −7.60239007294527072421775172501, −7.05374296981909562169259099530, −6.66960195184789502623846379940, −6.28244954984534037656450972334, −5.58273797500418171414855122983, −5.08744897014392907333183377880, −4.31455536593277949273106493230, −4.15194945563185952972528092753, −3.72489394124566965564657676313, −3.10905144406399840643454618813, −2.46473763190113210445772453261, −1.44107643273385105055050516054, −0.60155926564354526102394148198, 0.60155926564354526102394148198, 1.44107643273385105055050516054, 2.46473763190113210445772453261, 3.10905144406399840643454618813, 3.72489394124566965564657676313, 4.15194945563185952972528092753, 4.31455536593277949273106493230, 5.08744897014392907333183377880, 5.58273797500418171414855122983, 6.28244954984534037656450972334, 6.66960195184789502623846379940, 7.05374296981909562169259099530, 7.60239007294527072421775172501, 7.87524926693289186748273169556, 8.343180798202676655298184263414

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