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Properties

Label 4-338688-1.1-c1e2-0-19
Degree $4$
Conductor $338688$
Sign $1$
Analytic cond. $21.5950$
Root an. cond. $2.15570$
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  + 3-s + 2·7-s + 9-s − 12·13-s + 8·19-s + 2·21-s + 6·25-s + 27-s + 4·37-s − 12·39-s + 8·43-s + 3·49-s + 8·57-s + 12·61-s + 2·63-s + 16·67-s − 4·73-s + 6·75-s − 24·79-s + 81-s − 24·91-s − 4·97-s + 32·103-s − 4·109-s + 4·111-s − 12·117-s − 18·121-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s − 3.32·13-s + 1.83·19-s + 0.436·21-s + 6/5·25-s + 0.192·27-s + 0.657·37-s − 1.92·39-s + 1.21·43-s + 3/7·49-s + 1.05·57-s + 1.53·61-s + 0.251·63-s + 1.95·67-s − 0.468·73-s + 0.692·75-s − 2.70·79-s + 1/9·81-s − 2.51·91-s − 0.406·97-s + 3.15·103-s − 0.383·109-s + 0.379·111-s − 1.10·117-s − 1.63·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(338688\)    =    \(2^{8} \cdot 3^{3} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(21.5950\)
Root analytic conductor: \(2.15570\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 338688,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.207397655\)
\(L(\frac12)\) \(\approx\) \(2.207397655\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.13.m_ck
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.23.a_bq
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
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$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.71.a_acc
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.79.y_lq
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_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

−8.977835409861481770724787215218, −8.100150186340802045251146455502, −7.76556317647643738947138137602, −7.47655671664491235528608994140, −6.98585917665527551317684669265, −6.76281206095777722703467003614, −5.54203834680246413327649010589, −5.44643327053124053530313487020, −4.73251701657473593050786657983, −4.59055067570292347348864981302, −3.76740235739525618511594493326, −2.75718031114594033742569839966, −2.73919662589933967809346607483, −1.92538980972810959214451574073, −0.831916978658255590031556900046, 0.831916978658255590031556900046, 1.92538980972810959214451574073, 2.73919662589933967809346607483, 2.75718031114594033742569839966, 3.76740235739525618511594493326, 4.59055067570292347348864981302, 4.73251701657473593050786657983, 5.44643327053124053530313487020, 5.54203834680246413327649010589, 6.76281206095777722703467003614, 6.98585917665527551317684669265, 7.47655671664491235528608994140, 7.76556317647643738947138137602, 8.100150186340802045251146455502, 8.977835409861481770724787215218

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