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Properties

Label 4-2376e2-1.1-c1e2-0-13
Degree $4$
Conductor $5645376$
Sign $1$
Analytic cond. $359.954$
Root an. cond. $4.35573$
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  − 8·5-s − 4·11-s − 8·23-s + 38·25-s − 8·31-s − 18·37-s + 24·47-s − 5·49-s + 16·53-s + 32·55-s − 8·59-s + 22·67-s − 16·71-s − 24·89-s + 10·97-s + 2·103-s − 24·113-s + 64·115-s + 5·121-s − 136·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 64·155-s + ⋯
L(s)  = 1  − 3.57·5-s − 1.20·11-s − 1.66·23-s + 38/5·25-s − 1.43·31-s − 2.95·37-s + 3.50·47-s − 5/7·49-s + 2.19·53-s + 4.31·55-s − 1.04·59-s + 2.68·67-s − 1.89·71-s − 2.54·89-s + 1.01·97-s + 0.197·103-s − 2.25·113-s + 5.96·115-s + 5/11·121-s − 12.1·125-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 + 5.14·155-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: no
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$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.5.i_ba
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.a_z
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.a_bl
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.37.s_fz
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.47.ay_je
53$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.53.aq_go
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.61.a_dt
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.67.aw_jv
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.73.a_fp
79$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.79.a_fd
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.83.a_dy
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.89.y_mk
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.97.ak_il
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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

−6.95550481778817521461633350929, −6.93075982000562056071935732627, −5.98169863058984240039929554681, −5.48459428248370426498378218555, −5.16761096456984954836245442455, −4.69178179475877015203781084126, −3.99250331289836498435902686175, −3.96888445081768568889397071606, −3.73519918195197314997392920657, −3.09788365205681643264055516753, −2.65117332689213995396045885303, −1.93826743358928134231516571874, −0.887032512301407325004527560286, 0, 0, 0.887032512301407325004527560286, 1.93826743358928134231516571874, 2.65117332689213995396045885303, 3.09788365205681643264055516753, 3.73519918195197314997392920657, 3.96888445081768568889397071606, 3.99250331289836498435902686175, 4.69178179475877015203781084126, 5.16761096456984954836245442455, 5.48459428248370426498378218555, 5.98169863058984240039929554681, 6.93075982000562056071935732627, 6.95550481778817521461633350929

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