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Properties

Label 4-792e2-1.1-c1e2-0-24
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  + 2·2-s + 2·4-s − 6·11-s − 4·16-s − 6·17-s − 12·22-s + 7·25-s + 6·31-s − 8·32-s − 12·34-s + 12·37-s + 10·41-s − 12·44-s − 6·49-s + 14·50-s + 12·62-s − 8·64-s + 4·67-s − 12·68-s + 24·74-s + 20·82-s − 4·83-s + 6·97-s − 12·98-s + 14·100-s + 2·101-s + 24·103-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 1.80·11-s − 16-s − 1.45·17-s − 2.55·22-s + 7/5·25-s + 1.07·31-s − 1.41·32-s − 2.05·34-s + 1.97·37-s + 1.56·41-s − 1.80·44-s − 6/7·49-s + 1.97·50-s + 1.52·62-s − 64-s + 0.488·67-s − 1.45·68-s + 2.78·74-s + 2.20·82-s − 0.439·83-s + 0.609·97-s − 1.21·98-s + 7/5·100-s + 0.199·101-s + 2.36·103-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

\(L(1)\) \(\approx\) \(2.988390198\)
\(L(\frac12)\) \(\approx\) \(2.988390198\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
3 \( 1 \)
11$C_2$ \( 1 + 6 T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.5.a_ah
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
13$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.13.a_as
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.g_bq
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.19.a_ag
23$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.23.a_d
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.31.ag_cd
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.37.am_dh
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.41.ak_ec
43$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.43.a_abe
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.47.a_o
53$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.53.a_bi
59$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \) 2.59.a_cn
61$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.61.a_cc
67$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.67.ae_ez
71$C_2^2$ \( 1 + 79 T^{2} + p^{2} T^{4} \) 2.71.a_db
73$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.73.a_bq
79$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.79.a_abe
83$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.83.e_ec
89$C_2^2$ \( 1 + 51 T^{2} + p^{2} T^{4} \) 2.89.a_bz
97$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.97.ag_fj
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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

−8.343449140885439645656577311193, −7.78970742414657926142503879069, −7.48132257992839336209348896948, −6.81597070518766970726978811770, −6.47161013671627445225146258908, −5.96790200966431922475620247674, −5.61693619485487660572887039303, −4.93584099329112098291633920021, −4.59131911040672231342081257782, −4.43983007355245845338930934626, −3.58975400660762895406640078327, −2.89182530872395182156480184715, −2.62425638466576633897352305402, −2.10396995081044244318216788271, −0.66556160700882718734927216886, 0.66556160700882718734927216886, 2.10396995081044244318216788271, 2.62425638466576633897352305402, 2.89182530872395182156480184715, 3.58975400660762895406640078327, 4.43983007355245845338930934626, 4.59131911040672231342081257782, 4.93584099329112098291633920021, 5.61693619485487660572887039303, 5.96790200966431922475620247674, 6.47161013671627445225146258908, 6.81597070518766970726978811770, 7.48132257992839336209348896948, 7.78970742414657926142503879069, 8.343449140885439645656577311193

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