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Properties

Label 4-416e2-1.1-c1e2-0-24
Degree $4$
Conductor $173056$
Sign $1$
Analytic cond. $11.0342$
Root an. cond. $1.82257$
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  + 6·3-s + 21·9-s − 4·13-s − 10·17-s + 12·23-s + 25-s + 54·27-s − 8·29-s − 24·39-s + 6·43-s + 13·49-s − 60·51-s − 4·53-s − 24·61-s + 72·69-s + 6·75-s − 12·79-s + 108·81-s − 48·87-s − 8·101-s + 12·103-s − 24·107-s − 28·113-s − 84·117-s + 6·121-s + 127-s + 36·129-s + ⋯
L(s)  = 1  + 3.46·3-s + 7·9-s − 1.10·13-s − 2.42·17-s + 2.50·23-s + 1/5·25-s + 10.3·27-s − 1.48·29-s − 3.84·39-s + 0.914·43-s + 13/7·49-s − 8.40·51-s − 0.549·53-s − 3.07·61-s + 8.66·69-s + 0.692·75-s − 1.35·79-s + 12·81-s − 5.14·87-s − 0.796·101-s + 1.18·103-s − 2.32·107-s − 2.63·113-s − 7.76·117-s + 6/11·121-s + 0.0887·127-s + 3.16·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(173056\)    =    \(2^{10} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(11.0342\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 173056,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.367393556\)
\(L(\frac12)\) \(\approx\) \(5.367393556\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )^{2} \) 2.3.ag_p
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.7.a_an
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.17.k_ch
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.29.i_cw
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.37.a_acn
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.41.a_ck
43$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.43.ag_dr
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \) 2.47.a_abt
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.59.a_aek
61$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.61.y_kg
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \) 2.71.a_av
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.79.m_hm
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.83.a_aco
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.89.a_afm
97$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.97.a_ahm
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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

−11.09384862567865755762242940012, −10.95490282940223626651921197771, −10.42290777616891555720487265372, −9.624447414282299574499913141092, −9.275158666179834650021843116462, −9.249126006638488870170893159068, −8.628534650208570058426005696850, −8.597046724658197730158797259019, −7.67974329853721712331462689738, −7.53858091598994002164121181958, −6.92790219214488278954425235552, −6.86688662446558195187970104126, −5.66865797437829282466786215739, −4.56383055742702776520145173263, −4.53692360329425914856358761973, −3.83117076105982150257355341706, −3.06055133198430618695384486711, −2.76780131540227762149157645769, −2.24991262481566543765332430106, −1.61544043627670167664066694350, 1.61544043627670167664066694350, 2.24991262481566543765332430106, 2.76780131540227762149157645769, 3.06055133198430618695384486711, 3.83117076105982150257355341706, 4.53692360329425914856358761973, 4.56383055742702776520145173263, 5.66865797437829282466786215739, 6.86688662446558195187970104126, 6.92790219214488278954425235552, 7.53858091598994002164121181958, 7.67974329853721712331462689738, 8.597046724658197730158797259019, 8.628534650208570058426005696850, 9.249126006638488870170893159068, 9.275158666179834650021843116462, 9.624447414282299574499913141092, 10.42290777616891555720487265372, 10.95490282940223626651921197771, 11.09384862567865755762242940012

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