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Properties

Label 4-700e2-1.1-c1e2-0-14
Degree $4$
Conductor $490000$
Sign $1$
Analytic cond. $31.2428$
Root an. cond. $2.36421$
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  − 2·4-s − 5·9-s − 10·13-s + 4·16-s − 6·17-s + 6·29-s + 10·36-s − 4·37-s − 24·41-s + 49-s + 20·52-s − 24·53-s + 16·61-s − 8·64-s + 12·68-s − 4·73-s + 16·81-s − 24·89-s + 2·97-s + 12·101-s − 14·109-s − 12·113-s − 12·116-s + 50·117-s − 13·121-s + 127-s + 131-s + ⋯
L(s)  = 1  − 4-s − 5/3·9-s − 2.77·13-s + 16-s − 1.45·17-s + 1.11·29-s + 5/3·36-s − 0.657·37-s − 3.74·41-s + 1/7·49-s + 2.77·52-s − 3.29·53-s + 2.04·61-s − 64-s + 1.45·68-s − 0.468·73-s + 16/9·81-s − 2.54·89-s + 0.203·97-s + 1.19·101-s − 1.34·109-s − 1.12·113-s − 1.11·116-s + 4.62·117-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(490000\)    =    \(2^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.2428\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 490000,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + p T^{2} \)
5 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.13.k_bz
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.29.ag_cp
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.41.y_is
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.53.y_jq
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.79.a_gb
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.89.y_mk
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.97.ac_hn
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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

−8.214592059880155258547933248387, −7.86011514048466165442018696861, −6.93454011637102289378074371468, −6.88150838864812305101444716237, −6.27248387518076710770064262455, −5.45020178597402395728815947142, −5.19532839461654101726843493087, −4.78686073935120916368506592009, −4.46535279179624622242920849885, −3.55022395369242424556181791228, −2.98704088538274633622207409794, −2.54455347664810529993661521980, −1.77645134008769781761461177809, 0, 0, 1.77645134008769781761461177809, 2.54455347664810529993661521980, 2.98704088538274633622207409794, 3.55022395369242424556181791228, 4.46535279179624622242920849885, 4.78686073935120916368506592009, 5.19532839461654101726843493087, 5.45020178597402395728815947142, 6.27248387518076710770064262455, 6.88150838864812305101444716237, 6.93454011637102289378074371468, 7.86011514048466165442018696861, 8.214592059880155258547933248387

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