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Properties

Label 4-792e2-1.1-c1e2-0-35
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 − 2·11-s − 4·16-s + 4·17-s − 6·19-s − 4·22-s + 6·25-s − 8·32-s + 8·34-s − 12·38-s + 4·41-s − 6·43-s − 4·44-s + 2·49-s + 12·50-s + 8·59-s − 8·64-s + 16·67-s + 8·68-s + 20·73-s − 12·76-s + 8·82-s − 12·86-s + 18·89-s + 4·97-s + 4·98-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.603·11-s − 16-s + 0.970·17-s − 1.37·19-s − 0.852·22-s + 6/5·25-s − 1.41·32-s + 1.37·34-s − 1.94·38-s + 0.624·41-s − 0.914·43-s − 0.603·44-s + 2/7·49-s + 1.69·50-s + 1.04·59-s − 64-s + 1.95·67-s + 0.970·68-s + 2.34·73-s − 1.37·76-s + 0.883·82-s − 1.29·86-s + 1.90·89-s + 0.406·97-s + 0.404·98-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

\(L(1)\) \(\approx\) \(3.604548099\)
\(L(\frac12)\) \(\approx\) \(3.604548099\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.ae_c
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.g_bm
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.37.a_c
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.g_di
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.a_ag
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.a_abm
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.61.a_adq
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.67.aq_gg
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.79.a_acw
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.89.as_jq
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_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

−8.445636208819054692369028240041, −7.84916007498563660047536635960, −7.41242099124438021334479997520, −6.71754597367136890917087845876, −6.56616669516481251036150077540, −6.04309762679119616046490997407, −5.39268188712630011795781335023, −5.19026750134360559717433846293, −4.68204345157445926411652040057, −4.14761586270770494020252094484, −3.57647021144405895784857564306, −3.19972028796176887216562390655, −2.43701813971856651479317882998, −2.05253171243738248999995455722, −0.75189929987229785960152446342, 0.75189929987229785960152446342, 2.05253171243738248999995455722, 2.43701813971856651479317882998, 3.19972028796176887216562390655, 3.57647021144405895784857564306, 4.14761586270770494020252094484, 4.68204345157445926411652040057, 5.19026750134360559717433846293, 5.39268188712630011795781335023, 6.04309762679119616046490997407, 6.56616669516481251036150077540, 6.71754597367136890917087845876, 7.41242099124438021334479997520, 7.84916007498563660047536635960, 8.445636208819054692369028240041

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