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Properties

Label 4-2376e2-1.1-c1e2-0-9
Degree $4$
Conductor $5645376$
Sign $-1$
Analytic cond. $359.954$
Root an. cond. $4.35573$
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  − 5-s − 5·11-s + 6·23-s − 3·25-s − 7·31-s − 10·37-s + 7·47-s + 49-s + 8·53-s + 5·55-s − 59-s + 9·67-s + 12·71-s + 8·89-s − 97-s − 5·103-s + 17·113-s − 6·115-s + 14·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 7·155-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.50·11-s + 1.25·23-s − 3/5·25-s − 1.25·31-s − 1.64·37-s + 1.02·47-s + 1/7·49-s + 1.09·53-s + 0.674·55-s − 0.130·59-s + 1.09·67-s + 1.42·71-s + 0.847·89-s − 0.101·97-s − 0.492·103-s + 1.59·113-s − 0.559·115-s + 1.27·121-s + 0.178·125-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 + 0.562·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5645376\)    =    \(2^{6} \cdot 3^{6} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(359.954\)
Root analytic conductor: \(4.35573\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 5645376,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.b_e
7$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.7.a_ab
13$C_2^2$ \( 1 + 21 T^{2} + p^{2} T^{4} \) 2.13.a_v
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_g
23$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.ag_t
29$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.29.a_an
31$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.h_cc
37$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.k_dr
41$C_2^2$ \( 1 + 67 T^{2} + p^{2} T^{4} \) 2.41.a_cp
43$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.43.a_l
47$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.ah_dq
53$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.53.ai_ej
59$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.b_ec
61$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \) 2.61.a_abf
67$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.aj_de
71$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.71.am_ez
73$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.73.a_ba
79$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \) 2.79.a_av
83$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \) 2.83.a_j
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.89.ai_hi
97$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.b_hk
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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

−7.16633413531286368640809197621, −6.80077602362668751835955768063, −6.24532243860152612139304346595, −5.72903647953847466387316425754, −5.37102962094917612787851385674, −5.05469335715386464094979923951, −4.74967620451416255578841827900, −4.01758469059220763645795253920, −3.61700667181298144495509723460, −3.34611300254910108342742803322, −2.53221676589038407628878917411, −2.35827447095119636847007357983, −1.62062139942700753657933883752, −0.76577592668479485330664047736, 0, 0.76577592668479485330664047736, 1.62062139942700753657933883752, 2.35827447095119636847007357983, 2.53221676589038407628878917411, 3.34611300254910108342742803322, 3.61700667181298144495509723460, 4.01758469059220763645795253920, 4.74967620451416255578841827900, 5.05469335715386464094979923951, 5.37102962094917612787851385674, 5.72903647953847466387316425754, 6.24532243860152612139304346595, 6.80077602362668751835955768063, 7.16633413531286368640809197621

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