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Properties

Label 4-294e2-1.1-c1e2-0-6
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4-s + 9-s + 2·12-s + 8·13-s + 16-s − 4·19-s − 10·25-s − 4·27-s + 8·31-s + 36-s + 4·37-s + 16·39-s + 16·43-s + 2·48-s + 8·52-s − 8·57-s − 16·61-s + 64-s − 8·67-s − 4·73-s − 20·75-s − 4·76-s + 16·79-s − 11·81-s + 16·93-s + 20·97-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s + 1/3·9-s + 0.577·12-s + 2.21·13-s + 1/4·16-s − 0.917·19-s − 2·25-s − 0.769·27-s + 1.43·31-s + 1/6·36-s + 0.657·37-s + 2.56·39-s + 2.43·43-s + 0.288·48-s + 1.10·52-s − 1.05·57-s − 2.04·61-s + 1/8·64-s − 0.977·67-s − 0.468·73-s − 2.30·75-s − 0.458·76-s + 1.80·79-s − 1.22·81-s + 1.65·93-s + 2.03·97-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: \(0\)
Selberg data: \((4,\ 86436,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.599314781\)
\(L(\frac12)\) \(\approx\) \(2.599314781\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
7 \( 1 \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
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.ae_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_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

−9.393998538973797562285424645673, −9.205467104844532229657218112988, −8.624667079434487291694904422005, −8.245010328271296459357780366393, −7.64010214491199898083268392854, −7.55563465371540871894331307120, −6.37349457460642388728354325328, −6.12986038892916091786195943044, −5.88407913223931048395780461484, −4.74683922157615295493069282209, −3.87985229263682105648190826671, −3.79391663707115759548696099652, −2.83833445427180997893704571146, −2.25140513775369021989127014661, −1.33827299261372059142982522511, 1.33827299261372059142982522511, 2.25140513775369021989127014661, 2.83833445427180997893704571146, 3.79391663707115759548696099652, 3.87985229263682105648190826671, 4.74683922157615295493069282209, 5.88407913223931048395780461484, 6.12986038892916091786195943044, 6.37349457460642388728354325328, 7.55563465371540871894331307120, 7.64010214491199898083268392854, 8.245010328271296459357780366393, 8.624667079434487291694904422005, 9.205467104844532229657218112988, 9.393998538973797562285424645673

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