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Properties

Label 4-2112e2-1.1-c1e2-0-29
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  − 3-s − 4·5-s − 2·9-s + 4·11-s + 4·15-s + 8·23-s + 3·25-s + 5·27-s − 6·31-s − 4·33-s − 6·37-s + 8·45-s − 6·49-s + 16·53-s − 16·55-s + 12·59-s + 14·67-s − 8·69-s − 24·71-s − 3·75-s + 81-s + 8·89-s + 6·93-s − 6·97-s − 8·99-s − 8·103-s + 6·111-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s − 2/3·9-s + 1.20·11-s + 1.03·15-s + 1.66·23-s + 3/5·25-s + 0.962·27-s − 1.07·31-s − 0.696·33-s − 0.986·37-s + 1.19·45-s − 6/7·49-s + 2.19·53-s − 2.15·55-s + 1.56·59-s + 1.71·67-s − 0.963·69-s − 2.84·71-s − 0.346·75-s + 1/9·81-s + 0.847·89-s + 0.622·93-s − 0.609·97-s − 0.804·99-s − 0.788·103-s + 0.569·111-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$C_2$ \( 1 + T + p T^{2} \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.e_n
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.19.a_as
23$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.23.ai_cb
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.31.g_ct
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.37.g_bv
41$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.41.a_be
43$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.43.a_acc
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.53.aq_gk
59$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.59.am_fx
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.67.ao_gx
71$C_2$$\times$$C_2$ \( ( 1 + 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.71.y_kr
73$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.73.a_co
79$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.79.a_ek
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.83.a_di
89$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.ai_z
97$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.97.g_hv
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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

−7.12296835689531828419172314023, −6.96242379547504162594632274208, −6.43426161112425427802355654585, −5.83390414876252617735354391325, −5.60651088142050010085064931397, −4.96957496202027952333388779709, −4.74054722527680440231335392003, −4.07570044462901154585855245816, −3.69884500518693020828640564269, −3.53729039829921536298000164003, −2.89016245040251153046287074325, −2.25514668711721411393870235934, −1.39585341738655484686323839368, −0.73556239424125021006876266680, 0, 0.73556239424125021006876266680, 1.39585341738655484686323839368, 2.25514668711721411393870235934, 2.89016245040251153046287074325, 3.53729039829921536298000164003, 3.69884500518693020828640564269, 4.07570044462901154585855245816, 4.74054722527680440231335392003, 4.96957496202027952333388779709, 5.60651088142050010085064931397, 5.83390414876252617735354391325, 6.43426161112425427802355654585, 6.96242379547504162594632274208, 7.12296835689531828419172314023

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