Properties

Label 4-1536e2-1.1-c1e2-0-5
Degree $4$
Conductor $2359296$
Sign $1$
Analytic cond. $150.430$
Root an. cond. $3.50214$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s + 12·11-s − 12·17-s + 8·19-s − 8·25-s − 4·27-s − 24·33-s − 4·41-s + 4·49-s + 24·51-s − 16·57-s + 8·59-s + 16·67-s + 16·73-s + 16·75-s + 5·81-s + 4·83-s + 4·89-s + 4·97-s + 36·99-s − 8·107-s − 20·113-s + 86·121-s + 8·123-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s + 3.61·11-s − 2.91·17-s + 1.83·19-s − 8/5·25-s − 0.769·27-s − 4.17·33-s − 0.624·41-s + 4/7·49-s + 3.36·51-s − 2.11·57-s + 1.04·59-s + 1.95·67-s + 1.87·73-s + 1.84·75-s + 5/9·81-s + 0.439·83-s + 0.423·89-s + 0.406·97-s + 3.61·99-s − 0.773·107-s − 1.88·113-s + 7.81·121-s + 0.721·123-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2359296\)    =    \(2^{18} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(150.430\)
Root analytic conductor: \(3.50214\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2359296,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.729963195\)
\(L(\frac12)\) \(\approx\) \(1.729963195\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.5.a_i
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.7.a_ae
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.11.am_cg
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.23.a_bm
29$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \) 2.29.a_ce
31$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \) 2.31.a_ci
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.37.a_c
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.47.a_di
53$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.53.a_i
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.61.a_by
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \) 2.71.a_fe
73$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.73.aq_ic
79$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.79.a_ae
83$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.83.ae_go
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.89.ae_ha
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−9.479147082819314286887374233082, −9.329028849130003149968887434696, −8.974363245026290771882010558512, −8.729806734291771044247742700167, −7.969595087606173772397230191303, −7.58929421185305713271870038264, −6.82330659221935284042956344159, −6.76922470237093041071592885437, −6.49327045770963959786689606839, −6.25582842784896949155701324695, −5.36855720358405827873265334204, −5.36322584255679373857777577392, −4.45814122226574365449070356385, −4.25361943037059161249668463097, −3.70383039227769137582396865576, −3.63660714636738948326101316675, −2.40912332359023244618515680324, −1.84344958307031385167369536521, −1.30590714772851114620463181039, −0.61971493429863042851907788793, 0.61971493429863042851907788793, 1.30590714772851114620463181039, 1.84344958307031385167369536521, 2.40912332359023244618515680324, 3.63660714636738948326101316675, 3.70383039227769137582396865576, 4.25361943037059161249668463097, 4.45814122226574365449070356385, 5.36322584255679373857777577392, 5.36855720358405827873265334204, 6.25582842784896949155701324695, 6.49327045770963959786689606839, 6.76922470237093041071592885437, 6.82330659221935284042956344159, 7.58929421185305713271870038264, 7.969595087606173772397230191303, 8.729806734291771044247742700167, 8.974363245026290771882010558512, 9.329028849130003149968887434696, 9.479147082819314286887374233082

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