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Properties

Label 4-294e2-1.1-c1e2-0-9
Degree $4$
Conductor $86436$
Sign $-1$
Analytic cond. $5.51123$
Root an. cond. $1.53218$
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  − 2·2-s + 3·4-s − 4·8-s + 9-s − 8·11-s + 5·16-s − 2·18-s + 16·22-s + 6·25-s + 4·29-s − 6·32-s + 3·36-s − 12·37-s + 8·43-s − 24·44-s − 12·50-s − 20·53-s − 8·58-s + 7·64-s + 8·67-s + 16·71-s − 4·72-s + 24·74-s − 16·79-s + 81-s − 16·86-s + 32·88-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s + 1/3·9-s − 2.41·11-s + 5/4·16-s − 0.471·18-s + 3.41·22-s + 6/5·25-s + 0.742·29-s − 1.06·32-s + 1/2·36-s − 1.97·37-s + 1.21·43-s − 3.61·44-s − 1.69·50-s − 2.74·53-s − 1.05·58-s + 7/8·64-s + 0.977·67-s + 1.89·71-s − 0.471·72-s + 2.78·74-s − 1.80·79-s + 1/9·81-s − 1.72·86-s + 3.41·88-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86436\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4}\)
Sign: $-1$
Analytic conductor: \(5.51123\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 86436,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.53.u_hy
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.61.a_ec
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.89.a_ek
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.a_fa
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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

−9.234982336494788048482584951326, −9.128932118536250132875092109626, −8.236567804687403362610493554923, −8.056400623307585390849169627091, −7.69885140727101440087412366746, −6.94059953288016050503047892713, −6.70278112570871774442796552423, −5.90616655364322511705675042513, −5.07869549160647179585211788494, −5.02209723965846660973920127041, −3.77980943086478052006597909549, −2.83099639583900330764391450302, −2.52889971936759396667083662829, −1.40906194861780189487468921147, 0, 1.40906194861780189487468921147, 2.52889971936759396667083662829, 2.83099639583900330764391450302, 3.77980943086478052006597909549, 5.02209723965846660973920127041, 5.07869549160647179585211788494, 5.90616655364322511705675042513, 6.70278112570871774442796552423, 6.94059953288016050503047892713, 7.69885140727101440087412366746, 8.056400623307585390849169627091, 8.236567804687403362610493554923, 9.128932118536250132875092109626, 9.234982336494788048482584951326

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