Properties

Label 4-544e2-1.1-c1e2-0-20
Degree $4$
Conductor $295936$
Sign $-1$
Analytic cond. $18.8691$
Root an. cond. $2.08419$
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  − 6·9-s + 4·13-s − 2·17-s − 10·25-s + 16·29-s + 8·37-s − 4·41-s − 10·49-s − 12·53-s + 8·61-s − 12·73-s + 27·81-s − 20·89-s − 20·97-s + 4·101-s + 20·113-s − 24·117-s − 6·121-s + ⋯
L(s)  = 1  − 2·9-s + 1.10·13-s − 0.485·17-s − 2·25-s + 2.97·29-s + 1.31·37-s − 0.624·41-s − 1.42·49-s − 1.64·53-s + 1.02·61-s − 1.40·73-s + 3·81-s − 2.11·89-s − 2.03·97-s + 0.398·101-s + 1.88·113-s − 2.21·117-s − 0.545·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 295936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 295936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(295936\)    =    \(2^{10} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(18.8691\)
Root analytic conductor: \(2.08419\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 295936,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
17$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.29.aq_es
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.a_cg
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.61.ai_fi
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−8.407778739743522637911712748765, −8.325374186970197406187137193906, −7.949465060430367136751257620846, −7.19453824680681160758458923269, −6.45486383530876632990023364782, −6.18546001598565091594755313788, −5.92602088890819392149162970339, −5.24537194788322556785004984795, −4.68811629121503471986666453564, −4.10696532027485733787761686087, −3.37777254090361242162912900058, −2.89788769220777054973227426733, −2.34190413882648615913196631793, −1.28938575374331460272099629230, 0, 1.28938575374331460272099629230, 2.34190413882648615913196631793, 2.89788769220777054973227426733, 3.37777254090361242162912900058, 4.10696532027485733787761686087, 4.68811629121503471986666453564, 5.24537194788322556785004984795, 5.92602088890819392149162970339, 6.18546001598565091594755313788, 6.45486383530876632990023364782, 7.19453824680681160758458923269, 7.949465060430367136751257620846, 8.325374186970197406187137193906, 8.407778739743522637911712748765

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