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Properties

Label 4-792e2-1.1-c1e2-0-111
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 2·11-s − 6·25-s − 16·31-s − 4·37-s + 16·47-s − 2·49-s + 2·53-s − 4·55-s + 20·59-s + 12·67-s − 24·71-s − 10·89-s − 4·97-s + 4·103-s − 10·113-s − 7·121-s − 22·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 32·155-s + 157-s + 163-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.603·11-s − 6/5·25-s − 2.87·31-s − 0.657·37-s + 2.33·47-s − 2/7·49-s + 0.274·53-s − 0.539·55-s + 2.60·59-s + 1.46·67-s − 2.84·71-s − 1.05·89-s − 0.406·97-s + 0.394·103-s − 0.940·113-s − 0.636·121-s − 1.96·125-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 + ⋯

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: \(1\)
Selberg data: \((4,\ 627264,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.5.ac_k
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.17.a_ak
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.19.a_ag
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.29.a_abi
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.e_ck
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.41.a_ac
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.a_by
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.47.aq_fm
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.ac_du
59$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.59.au_hu
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.61.a_ak
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.67.am_gk
71$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.y_kk
73$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.73.a_adq
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$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.k_hu
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.e_cc
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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

−8.208698350501136865822409587022, −7.60098898575946103655366101396, −7.23196772200456881961471274007, −6.97437474611938562753462394133, −6.20515810366943087228049968621, −5.75156434949229095510045245007, −5.41893256682065107138105026687, −5.21532340665130469673283152359, −4.21729022605330335351110523059, −3.90223284501489060439609042327, −3.30850890224101991277644795477, −2.38384924502343313739920076963, −2.15663829479884840942590429087, −1.33949072842649607600712159481, 0, 1.33949072842649607600712159481, 2.15663829479884840942590429087, 2.38384924502343313739920076963, 3.30850890224101991277644795477, 3.90223284501489060439609042327, 4.21729022605330335351110523059, 5.21532340665130469673283152359, 5.41893256682065107138105026687, 5.75156434949229095510045245007, 6.20515810366943087228049968621, 6.97437474611938562753462394133, 7.23196772200456881961471274007, 7.60098898575946103655366101396, 8.208698350501136865822409587022

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