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Properties

Label 4-2376e2-1.1-c1e2-0-21
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 11-s + 9·23-s + 3·25-s − 13·31-s − 10·37-s − 11·47-s + 49-s − 16·53-s − 4·55-s + 17·59-s − 24·67-s − 12·71-s − 16·89-s − 25·97-s + 7·103-s − 19·113-s − 36·115-s − 10·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 52·155-s + ⋯
L(s)  = 1  − 1.78·5-s + 0.301·11-s + 1.87·23-s + 3/5·25-s − 2.33·31-s − 1.64·37-s − 1.60·47-s + 1/7·49-s − 2.19·53-s − 0.539·55-s + 2.21·59-s − 2.93·67-s − 1.42·71-s − 1.69·89-s − 2.53·97-s + 0.689·103-s − 1.78·113-s − 3.35·115-s − 0.909·121-s + 1.78·125-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 + 4.17·155-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: \(2\)
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$C_2$ \( 1 - T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.e_n
7$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.7.a_ab
13$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \) 2.13.a_j
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.19.a_as
23$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) 2.23.aj_bu
29$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.29.a_az
31$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.n_dy
37$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.k_dr
41$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.41.a_h
43$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.43.a_l
47$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.l_eo
53$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.53.q_gf
59$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.59.ar_hi
61$C_2^2$ \( 1 + 77 T^{2} + p^{2} T^{4} \) 2.61.a_cz
67$C_2$$\times$$C_2$ \( ( 1 + 11 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.67.y_kr
71$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.71.m_gv
73$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.73.a_acs
79$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \) 2.79.a_abh
83$C_2^2$ \( 1 + 57 T^{2} + p^{2} T^{4} \) 2.83.a_cf
89$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.q_jh
97$C_2$$\times$$C_2$ \( ( 1 + 11 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.z_nk
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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

−6.99008351112144867828132726778, −6.71009712303216139987961119725, −6.01840165486712256394410566071, −5.59459766181063754557225259925, −5.12177799954845435110031833894, −4.80120884299018877105164886872, −4.25501257325885521198006811678, −3.86347903695325807383488654648, −3.49638273850540064066612353925, −3.13848265773012698189545314578, −2.61977503688896519936001722528, −1.61620180006883912057534848888, −1.37949110119448656302256897283, 0, 0, 1.37949110119448656302256897283, 1.61620180006883912057534848888, 2.61977503688896519936001722528, 3.13848265773012698189545314578, 3.49638273850540064066612353925, 3.86347903695325807383488654648, 4.25501257325885521198006811678, 4.80120884299018877105164886872, 5.12177799954845435110031833894, 5.59459766181063754557225259925, 6.01840165486712256394410566071, 6.71009712303216139987961119725, 6.99008351112144867828132726778

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