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Properties

Label 4-704e2-1.1-c1e2-0-14
Degree $4$
Conductor $495616$
Sign $1$
Analytic cond. $31.6009$
Root an. cond. $2.37096$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s − 5·9-s + 17·25-s − 14·37-s + 30·45-s − 14·49-s − 12·53-s + 16·81-s − 18·89-s − 34·97-s − 42·113-s − 11·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 84·185-s + ⋯
L(s)  = 1  − 2.68·5-s − 5/3·9-s + 17/5·25-s − 2.30·37-s + 4.47·45-s − 2·49-s − 1.64·53-s + 16/9·81-s − 1.90·89-s − 3.45·97-s − 3.95·113-s − 121-s − 1.60·125-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 + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 6.17·185-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(495616\)    =    \(2^{12} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(31.6009\)
Root analytic conductor: \(2.37096\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 495616,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
11$C_2$ \( 1 + p T^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.17.a_abi
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.23.a_abj
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.a_bl
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.37.o_et
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.59.a_aed
61$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.61.a_aes
67$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.67.a_abj
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.71.a_fd
73$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.73.a_afq
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.89.s_jz
97$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \) 2.97.bi_sp
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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

−10.49894472254340161155107142913, −9.666790338973587515622523736203, −9.326312862081773032875803090117, −8.594361729608586267554812323925, −8.493572644889958533464086128990, −8.035784411417860809709606041602, −7.81490559442887454751120629388, −7.31493305562632928867511194204, −6.58221105065033544939748795416, −6.57670977616043792829215727312, −5.45762465322230015236764744954, −5.33641654692689145409582078387, −4.60097662653138778358261583646, −4.07743882859935420998233066208, −3.64305100300231984201598075692, −3.12698146552960151806648300996, −2.78435280141425654658403708045, −1.54249487062163749925713422612, 0, 0, 1.54249487062163749925713422612, 2.78435280141425654658403708045, 3.12698146552960151806648300996, 3.64305100300231984201598075692, 4.07743882859935420998233066208, 4.60097662653138778358261583646, 5.33641654692689145409582078387, 5.45762465322230015236764744954, 6.57670977616043792829215727312, 6.58221105065033544939748795416, 7.31493305562632928867511194204, 7.81490559442887454751120629388, 8.035784411417860809709606041602, 8.493572644889958533464086128990, 8.594361729608586267554812323925, 9.326312862081773032875803090117, 9.666790338973587515622523736203, 10.49894472254340161155107142913

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