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Properties

Label 4-2112e2-1.1-c1e2-0-48
Degree $4$
Conductor $4460544$
Sign $-1$
Analytic cond. $284.408$
Root an. cond. $4.10662$
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  + 6·5-s − 3·9-s + 18·25-s − 24·37-s − 18·45-s + 18·53-s + 9·81-s − 26·89-s − 16·97-s − 2·113-s − 11·121-s + 30·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s − 144·185-s + 191-s + ⋯
L(s)  = 1  + 2.68·5-s − 9-s + 18/5·25-s − 3.94·37-s − 2.68·45-s + 2.47·53-s + 81-s − 2.75·89-s − 1.62·97-s − 0.188·113-s − 121-s + 2.68·125-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s − 10.5·185-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4460544\)    =    \(2^{12} \cdot 3^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(284.408\)
Root analytic conductor: \(4.10662\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 4460544,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.5.ag_s
7$C_2^2$ \( 1 + p^{2} T^{4} \) 2.7.a_a
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.17.a_aq
19$C_2^2$ \( 1 + p^{2} T^{4} \) 2.19.a_a
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.29.a_bo
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.37.y_ik
41$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.41.a_dc
43$C_2^2$ \( 1 + p^{2} T^{4} \) 2.43.a_a
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.53.as_gg
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.a_aw
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2^2$ \( 1 + p^{2} T^{4} \) 2.79.a_a
83$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.83.a_agk
89$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.ba_na
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.97.q_jy
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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

−6.99278453499302210445509604902, −6.74413897647122094273358945091, −6.24571150569938989504985626830, −5.85406042084408810521191261043, −5.50747588663361145433100841879, −5.26317227480593859899687577786, −5.06043864040534603066248930542, −4.19602932798201297811110639503, −3.65780244418590098300416696678, −3.10177694734470180989366578927, −2.55193184751158702750844388805, −2.23206920637268849282455526533, −1.68784372256960308545661045043, −1.24416640992479258811079628782, 0, 1.24416640992479258811079628782, 1.68784372256960308545661045043, 2.23206920637268849282455526533, 2.55193184751158702750844388805, 3.10177694734470180989366578927, 3.65780244418590098300416696678, 4.19602932798201297811110639503, 5.06043864040534603066248930542, 5.26317227480593859899687577786, 5.50747588663361145433100841879, 5.85406042084408810521191261043, 6.24571150569938989504985626830, 6.74413897647122094273358945091, 6.99278453499302210445509604902

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