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Properties

Label 4-1769472-1.1-c1e2-0-4
Degree $4$
Conductor $1769472$
Sign $1$
Analytic cond. $112.823$
Root an. cond. $3.25911$
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 + 9-s − 2·17-s − 6·25-s − 27-s + 6·41-s + 12·43-s + 2·49-s + 2·51-s + 12·59-s + 4·73-s + 6·75-s + 81-s − 24·83-s + 2·89-s + 16·97-s + 12·107-s + 10·113-s − 18·121-s − 6·123-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.485·17-s − 6/5·25-s − 0.192·27-s + 0.937·41-s + 1.82·43-s + 2/7·49-s + 0.280·51-s + 1.56·59-s + 0.468·73-s + 0.692·75-s + 1/9·81-s − 2.63·83-s + 0.211·89-s + 1.62·97-s + 1.16·107-s + 0.940·113-s − 1.63·121-s − 0.541·123-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1769472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1769472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1769472\)    =    \(2^{16} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(112.823\)
Root analytic conductor: \(3.25911\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1769472,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.384647931\)
\(L(\frac12)\) \(\approx\) \(1.384647931\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.c_ba
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.29.a_o
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.31.a_ak
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.ag_de
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.43.am_di
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.53.a_bm
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.59.am_fu
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.a_aw
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.71.a_bi
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.ae_g
79$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.79.a_bu
83$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.83.y_ko
89$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.ac_abu
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.aq_io
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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

−7.69677343264359076907086645239, −7.37090198712564915953952341765, −7.04818052686521246512510742453, −6.41468632442556430872847575918, −6.11920231260583850771607073300, −5.65314750967623737254615441403, −5.35652944197629677534728552660, −4.70594464677011783990183130001, −4.24067695039734165159934249303, −3.93590342565513988579310929255, −3.32915147487782663641802590418, −2.51910788510081903426378707443, −2.19512080775744317616567425367, −1.33972859989317328191995547158, −0.53391519101082852004445527702, 0.53391519101082852004445527702, 1.33972859989317328191995547158, 2.19512080775744317616567425367, 2.51910788510081903426378707443, 3.32915147487782663641802590418, 3.93590342565513988579310929255, 4.24067695039734165159934249303, 4.70594464677011783990183130001, 5.35652944197629677534728552660, 5.65314750967623737254615441403, 6.11920231260583850771607073300, 6.41468632442556430872847575918, 7.04818052686521246512510742453, 7.37090198712564915953952341765, 7.69677343264359076907086645239

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