Properties

Label 4-4608e2-1.1-c1e2-0-10
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 2·13-s + 4·17-s + 2·25-s − 6·29-s + 10·37-s + 14·49-s − 18·53-s + 2·61-s + 4·65-s − 8·85-s − 16·97-s − 18·101-s − 14·109-s + 32·113-s − 10·125-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.554·13-s + 0.970·17-s + 2/5·25-s − 1.11·29-s + 1.64·37-s + 2·49-s − 2.47·53-s + 0.256·61-s + 0.496·65-s − 0.867·85-s − 1.62·97-s − 1.79·101-s − 1.34·109-s + 3.01·113-s − 0.894·125-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

\(L(1)\) \(\approx\) \(1.112615381\)
\(L(\frac12)\) \(\approx\) \(1.112615381\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.c_c
7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2^2$ \( 1 + p^{2} T^{4} \) 2.11.a_a
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.c_c
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2^2$ \( 1 + p^{2} T^{4} \) 2.19.a_a
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.g_s
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ak_by
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2^2$ \( 1 + p^{2} T^{4} \) 2.43.a_a
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.s_gg
59$C_2^2$ \( 1 + p^{2} T^{4} \) 2.59.a_a
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.ac_c
67$C_2^2$ \( 1 + p^{2} T^{4} \) 2.67.a_a
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 + p^{2} T^{4} \) 2.83.a_a
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.97.q_jy
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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

−8.408136774374696928707055548920, −8.135940626952641601173350115654, −7.68053301862154317087756838823, −7.40896050756064396496300993967, −7.31130137678888448329762101548, −6.73480970691054747738602985457, −6.33391879855798004712672143300, −5.83555090774764609627406411643, −5.69503678010246730908602220015, −5.09895939008118643802400314244, −4.80788401633432597973631454158, −4.33816437265730018753424059107, −3.96871517579914671425082583830, −3.60557553419431523933555408655, −3.19237068276622828451922026474, −2.61653574881988436452322548495, −2.37145897462708202412010607955, −1.53953823529426858577703303322, −1.10436837292597160376676856024, −0.32221439056384940038353460715, 0.32221439056384940038353460715, 1.10436837292597160376676856024, 1.53953823529426858577703303322, 2.37145897462708202412010607955, 2.61653574881988436452322548495, 3.19237068276622828451922026474, 3.60557553419431523933555408655, 3.96871517579914671425082583830, 4.33816437265730018753424059107, 4.80788401633432597973631454158, 5.09895939008118643802400314244, 5.69503678010246730908602220015, 5.83555090774764609627406411643, 6.33391879855798004712672143300, 6.73480970691054747738602985457, 7.31130137678888448329762101548, 7.40896050756064396496300993967, 7.68053301862154317087756838823, 8.135940626952641601173350115654, 8.408136774374696928707055548920

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