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Properties

Label 4-2057e2-1.1-c1e2-0-5
Degree $4$
Conductor $4231249$
Sign $-1$
Analytic cond. $269.788$
Root an. cond. $4.05280$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 4·4-s + 6·9-s − 16·12-s + 12·16-s + 12·23-s − 10·25-s − 4·27-s + 8·31-s − 24·36-s − 4·37-s + 6·47-s + 48·48-s − 5·49-s − 6·53-s + 18·59-s − 32·64-s − 10·67-s + 48·69-s − 24·71-s − 40·75-s − 37·81-s − 30·89-s − 48·92-s + 32·93-s + 16·97-s + 40·100-s + ⋯
L(s)  = 1  + 2.30·3-s − 2·4-s + 2·9-s − 4.61·12-s + 3·16-s + 2.50·23-s − 2·25-s − 0.769·27-s + 1.43·31-s − 4·36-s − 0.657·37-s + 0.875·47-s + 6.92·48-s − 5/7·49-s − 0.824·53-s + 2.34·59-s − 4·64-s − 1.22·67-s + 5.77·69-s − 2.84·71-s − 4.61·75-s − 4.11·81-s − 3.17·89-s − 5.00·92-s + 3.31·93-s + 1.62·97-s + 4·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4231249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4231249 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4231249\)    =    \(11^{4} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(269.788\)
Root analytic conductor: \(4.05280\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 4231249,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad11 \( 1 \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.2.a_e
3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.3.ae_k
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.a_ax
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.a_b
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.a_by
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.47.ag_dz
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.53.g_el
59$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.59.as_hr
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.a_di
67$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.67.k_gd
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.73.a_cn
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.89.be_pn
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.97.aq_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

−7.31772661569686145839950009537, −7.24608117046774542434607176007, −6.32422609820024788903427802150, −5.65570170042982434902737626376, −5.63552611516237010198355246364, −4.92780655818062801617168633088, −4.33690474543179484375177890543, −4.31819599976110427100296482240, −3.60796211674639736885273377886, −3.17120571829008549416473920677, −3.09552236851761980264613157620, −2.40527884395801481706315448413, −1.70245103388878777142529973135, −1.00837451956744911826605701714, 0, 1.00837451956744911826605701714, 1.70245103388878777142529973135, 2.40527884395801481706315448413, 3.09552236851761980264613157620, 3.17120571829008549416473920677, 3.60796211674639736885273377886, 4.31819599976110427100296482240, 4.33690474543179484375177890543, 4.92780655818062801617168633088, 5.63552611516237010198355246364, 5.65570170042982434902737626376, 6.32422609820024788903427802150, 7.24608117046774542434607176007, 7.31772661569686145839950009537

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