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Properties

Label 4-238e2-1.1-c1e2-0-1
Degree $4$
Conductor $56644$
Sign $1$
Analytic cond. $3.61167$
Root an. cond. $1.37856$
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-s − 4·9-s + 6·13-s + 16-s − 2·17-s + 6·19-s + 4·25-s − 4·36-s + 4·43-s − 4·47-s + 49-s + 6·52-s + 4·53-s − 6·59-s + 64-s + 8·67-s − 2·68-s + 6·76-s + 7·81-s + 22·83-s − 12·89-s + 4·100-s − 2·101-s − 12·103-s − 24·117-s + 4·121-s + 127-s + ⋯
L(s)  = 1  + 1/2·4-s − 4/3·9-s + 1.66·13-s + 1/4·16-s − 0.485·17-s + 1.37·19-s + 4/5·25-s − 2/3·36-s + 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.832·52-s + 0.549·53-s − 0.781·59-s + 1/8·64-s + 0.977·67-s − 0.242·68-s + 0.688·76-s + 7/9·81-s + 2.41·83-s − 1.27·89-s + 2/5·100-s − 0.199·101-s − 1.18·103-s − 2.21·117-s + 4/11·121-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(56644\)    =    \(2^{2} \cdot 7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(3.61167\)
Root analytic conductor: \(1.37856\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 56644,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.552871243\)
\(L(\frac12)\) \(\approx\) \(1.552871243\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
17$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
5$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.5.a_ae
11$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.11.a_ae
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.13.ag_bi
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.19.ag_bu
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.29.a_q
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.37.a_aq
41$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.41.a_co
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.ae_cc
47$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.e_ck
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.ae_bu
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.59.g_da
61$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.61.a_bc
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.67.ai_fe
71$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.71.a_di
73$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \) 2.73.a_ew
79$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.79.a_bm
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.83.aw_la
89$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.m_hq
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.97.a_adq
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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

−10.01369863191046656342482592276, −9.377784940033828757275953165273, −8.964498816962928487035486683896, −8.452125071981328787779137017901, −8.069518567281400568465291177138, −7.44346269170023091679455439481, −6.74533334945065280154032721833, −6.33636412935708458449486201354, −5.69733939940589390360289879977, −5.36365823075046758414969005610, −4.50012905256360902366657811456, −3.56197239650088456581969530402, −3.15975507943514566352418698191, −2.33544513817312020241594604877, −1.12456810610028248142942155042, 1.12456810610028248142942155042, 2.33544513817312020241594604877, 3.15975507943514566352418698191, 3.56197239650088456581969530402, 4.50012905256360902366657811456, 5.36365823075046758414969005610, 5.69733939940589390360289879977, 6.33636412935708458449486201354, 6.74533334945065280154032721833, 7.44346269170023091679455439481, 8.069518567281400568465291177138, 8.452125071981328787779137017901, 8.964498816962928487035486683896, 9.377784940033828757275953165273, 10.01369863191046656342482592276

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