Properties

Label 4-4608e2-1.1-c1e2-0-13
Degree $4$
Conductor $21233664$
Sign $1$
Analytic cond. $1353.87$
Root an. cond. $6.06589$
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  − 8·11-s + 12·13-s − 12·23-s + 10·25-s + 12·37-s + 12·47-s − 4·49-s + 8·59-s − 12·61-s + 12·71-s + 12·73-s − 32·83-s − 24·97-s − 32·107-s − 12·109-s + 26·121-s + ⋯
L(s)  = 1  − 2.41·11-s + 3.32·13-s − 2.50·23-s + 2·25-s + 1.97·37-s + 1.75·47-s − 4/7·49-s + 1.04·59-s − 1.53·61-s + 1.42·71-s + 1.40·73-s − 3.51·83-s − 2.43·97-s − 3.09·107-s − 1.14·109-s + 2.36·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21233664\)    =    \(2^{18} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1353.87\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21233664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.913015216\)
\(L(\frac12)\) \(\approx\) \(1.913015216\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.17.a_aq
19$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.19.a_abe
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.29.a_o
31$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.31.a_abs
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.41.a_adc
43$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.43.a_ada
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.53.a_abi
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.67.a_ag
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2^2$ \( 1 - 140 T^{2} + p^{2} T^{4} \) 2.79.a_afk
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.83.bg_qg
89$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.89.a_aq
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.97.y_na
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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

−8.482742391057392924453213092066, −8.112592891628905199740111002068, −8.090167440716650064250800758049, −7.43112221869978720322374709592, −7.15234453094245160111075735951, −6.57172976214980727550018344216, −6.12834950370707309368471174315, −6.02145864016419743520450380605, −5.71659592641417647181930744363, −5.07906657771041767026182038988, −5.05339912625417010124755306154, −4.08545751121639650796935692878, −4.04941845596881427924300363162, −3.74312796316993587308840107404, −2.97413500515241926631991146035, −2.60048077075518605376445313580, −2.46485288819017933362252737988, −1.40433944883755445226025262015, −1.27757698326604490686496290062, −0.40256190273376873416769096049, 0.40256190273376873416769096049, 1.27757698326604490686496290062, 1.40433944883755445226025262015, 2.46485288819017933362252737988, 2.60048077075518605376445313580, 2.97413500515241926631991146035, 3.74312796316993587308840107404, 4.04941845596881427924300363162, 4.08545751121639650796935692878, 5.05339912625417010124755306154, 5.07906657771041767026182038988, 5.71659592641417647181930744363, 6.02145864016419743520450380605, 6.12834950370707309368471174315, 6.57172976214980727550018344216, 7.15234453094245160111075735951, 7.43112221869978720322374709592, 8.090167440716650064250800758049, 8.112592891628905199740111002068, 8.482742391057392924453213092066

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