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Properties

Label 4-1568e2-1.1-c1e2-0-27
Degree $4$
Conductor $2458624$
Sign $1$
Analytic cond. $156.763$
Root an. cond. $3.53843$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s + 9-s − 8·13-s + 2·17-s + 17·25-s − 8·29-s − 10·37-s + 16·41-s − 6·45-s + 14·53-s − 10·61-s + 48·65-s − 18·73-s − 8·81-s − 12·85-s − 18·89-s − 16·97-s − 30·101-s + 2·109-s − 8·117-s − 15·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + ⋯
L(s)  = 1  − 2.68·5-s + 1/3·9-s − 2.21·13-s + 0.485·17-s + 17/5·25-s − 1.48·29-s − 1.64·37-s + 2.49·41-s − 0.894·45-s + 1.92·53-s − 1.28·61-s + 5.95·65-s − 2.10·73-s − 8/9·81-s − 1.30·85-s − 1.90·89-s − 1.62·97-s − 2.98·101-s + 0.191·109-s − 0.739·117-s − 1.36·121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2458624\)    =    \(2^{10} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(156.763\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2458624,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.3.a_ab
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
11$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.11.a_p
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.17.ac_bj
19$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.19.a_az
23$C_2^2$ \( 1 + 39 T^{2} + p^{2} T^{4} \) 2.23.a_bn
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.29.i_cw
31$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.31.a_cd
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.37.k_dv
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.41.aq_fq
43$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.43.a_aba
47$C_2^2$ \( 1 + 87 T^{2} + p^{2} T^{4} \) 2.47.a_dj
53$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.53.ao_fz
59$C_2^2$ \( 1 + 111 T^{2} + p^{2} T^{4} \) 2.59.a_eh
61$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.61.k_fr
67$C_2^2$ \( 1 + 127 T^{2} + p^{2} T^{4} \) 2.67.a_ex
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.73.s_it
79$C_2^2$ \( 1 + 151 T^{2} + p^{2} T^{4} \) 2.79.a_fv
83$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.83.a_cc
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.89.s_jz
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

−9.184103940456518015149208806706, −8.789854742685838728323248138339, −8.249449341935105524748386875809, −7.967405898665135453413551990507, −7.53913952804632410949447415316, −7.26726060947591664900913283788, −7.17670856704358346602985306090, −6.73831694020147380196659511550, −5.69234076361403692922051143065, −5.59166095201616983637751160015, −4.96773852082193860014089291001, −4.36750459662573376651309619949, −4.12657124781325397932046710664, −3.94526305368862862693663543105, −3.13105759066402359655423564997, −2.85034690542453184245271084515, −2.13249797555959133595684337921, −1.18726156001547987362434063658, 0, 0, 1.18726156001547987362434063658, 2.13249797555959133595684337921, 2.85034690542453184245271084515, 3.13105759066402359655423564997, 3.94526305368862862693663543105, 4.12657124781325397932046710664, 4.36750459662573376651309619949, 4.96773852082193860014089291001, 5.59166095201616983637751160015, 5.69234076361403692922051143065, 6.73831694020147380196659511550, 7.17670856704358346602985306090, 7.26726060947591664900913283788, 7.53913952804632410949447415316, 7.967405898665135453413551990507, 8.249449341935105524748386875809, 8.789854742685838728323248138339, 9.184103940456518015149208806706

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