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Properties

Label 4-2e18-1.1-c1e2-0-23
Degree $4$
Conductor $262144$
Sign $1$
Analytic cond. $16.7145$
Root an. cond. $2.02196$
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  − 4·5-s − 4·9-s − 12·13-s + 2·25-s − 4·29-s − 12·37-s + 12·41-s + 16·45-s − 6·49-s + 4·53-s − 12·61-s + 48·65-s − 24·73-s + 7·81-s − 24·89-s − 16·97-s + 20·101-s + 12·109-s + 12·113-s + 48·117-s − 4·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + ⋯
L(s)  = 1  − 1.78·5-s − 4/3·9-s − 3.32·13-s + 2/5·25-s − 0.742·29-s − 1.97·37-s + 1.87·41-s + 2.38·45-s − 6/7·49-s + 0.549·53-s − 1.53·61-s + 5.95·65-s − 2.80·73-s + 7/9·81-s − 2.54·89-s − 1.62·97-s + 1.99·101-s + 1.14·109-s + 1.12·113-s + 4.43·117-s − 0.363·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(262144\)    =    \(2^{18}\)
Sign: $1$
Analytic conductor: \(16.7145\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 262144,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.11.a_e
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.13.m_ck
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.19.a_u
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.23.a_aba
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.31.a_be
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \) 2.43.a_cq
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_eg
59$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \) 2.59.a_em
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \) 2.67.a_abc
71$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.71.a_cs
73$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.73.y_le
79$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \) 2.79.a_ew
83$C_2^2$ \( 1 + 148 T^{2} + p^{2} T^{4} \) 2.83.a_fs
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.89.y_mk
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

−10.72777274474817207367752876802, −10.28036655821767643575401024878, −9.635349597153063340889306226847, −9.513866428598459690325447452693, −8.687742184317362888336887425320, −8.518825838212268140126510294240, −7.74175324581225334869780225691, −7.55008140780084358241254172304, −7.29219538069229399258538049260, −6.80409189002055853973715740956, −5.70181878856779292233128223900, −5.69057393391727657466839067611, −4.69684119897158394604115836867, −4.62437680388923286668431169918, −3.87700045654179342788404540255, −3.15810655691968149432002287791, −2.73634755709990054979891173376, −2.00670986314658715682425526222, 0, 0, 2.00670986314658715682425526222, 2.73634755709990054979891173376, 3.15810655691968149432002287791, 3.87700045654179342788404540255, 4.62437680388923286668431169918, 4.69684119897158394604115836867, 5.69057393391727657466839067611, 5.70181878856779292233128223900, 6.80409189002055853973715740956, 7.29219538069229399258538049260, 7.55008140780084358241254172304, 7.74175324581225334869780225691, 8.518825838212268140126510294240, 8.687742184317362888336887425320, 9.513866428598459690325447452693, 9.635349597153063340889306226847, 10.28036655821767643575401024878, 10.72777274474817207367752876802

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