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Properties

Label 4-1520e2-1.1-c1e2-0-21
Degree $4$
Conductor $2310400$
Sign $1$
Analytic cond. $147.313$
Root an. cond. $3.48385$
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 − 5-s + 3·9-s + 6·13-s − 2·15-s + 6·17-s + 19-s + 10·27-s + 3·29-s − 10·31-s + 12·39-s + 12·41-s + 12·43-s − 3·45-s + 6·47-s + 14·49-s + 12·51-s + 6·53-s + 2·57-s + 9·59-s − 61-s − 6·65-s − 10·67-s − 15·71-s − 4·73-s + 79-s + 20·81-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 9-s + 1.66·13-s − 0.516·15-s + 1.45·17-s + 0.229·19-s + 1.92·27-s + 0.557·29-s − 1.79·31-s + 1.92·39-s + 1.87·41-s + 1.82·43-s − 0.447·45-s + 0.875·47-s + 2·49-s + 1.68·51-s + 0.824·53-s + 0.264·57-s + 1.17·59-s − 0.128·61-s − 0.744·65-s − 1.22·67-s − 1.78·71-s − 0.468·73-s + 0.112·79-s + 20/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2310400\)    =    \(2^{8} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(147.313\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2310400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.629621126\)
\(L(\frac12)\) \(\approx\) \(4.629621126\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
19$C_2$ \( 1 - T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_b
7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \) 2.11.a_at
13$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_z
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_t
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.23.a_x
29$C_2^2$ \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_bg
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.31.k_dj
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2^2$ \( 1 - 12 T + 89 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.41.am_dl
43$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.43.am_dn
47$C_2^2$ \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_ch
53$C_2^2$ \( 1 - 6 T + 65 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_cn
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.59.aj_w
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.b_aci
67$C_2^2$ \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.67.k_bh
71$C_2^2$ \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.71.p_fy
73$C_2^2$ \( 1 + 4 T - 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.73.e_acf
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_ada
83$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.83.a_aeo
89$C_2^2$ \( 1 - 21 T + 236 T^{2} - 21 p T^{3} + p^{2} T^{4} \) 2.89.av_jc
97$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.y_ld
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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

−9.320576061133271340883240942926, −9.249912150525896510589861719807, −8.907286448072998066641353006539, −8.383933919996533490564267593914, −8.283988971398823225397696768145, −7.50255694952984181539576145702, −7.35360239021729654192405019265, −7.28010364615867253685542716583, −6.21864751608379818851955167329, −6.19129092327315671496080255088, −5.38644793421509291664460148444, −5.36164290491031997884685696860, −4.22218636281074385259341468424, −4.09950441157193164190395932380, −3.79503044026715834797865539304, −3.10808541169506490228905237867, −2.76833914382719276988143747557, −2.19739626254719978384070032400, −1.17771923183519051941115026815, −1.00340578533254612489691958624, 1.00340578533254612489691958624, 1.17771923183519051941115026815, 2.19739626254719978384070032400, 2.76833914382719276988143747557, 3.10808541169506490228905237867, 3.79503044026715834797865539304, 4.09950441157193164190395932380, 4.22218636281074385259341468424, 5.36164290491031997884685696860, 5.38644793421509291664460148444, 6.19129092327315671496080255088, 6.21864751608379818851955167329, 7.28010364615867253685542716583, 7.35360239021729654192405019265, 7.50255694952984181539576145702, 8.283988971398823225397696768145, 8.383933919996533490564267593914, 8.907286448072998066641353006539, 9.249912150525896510589861719807, 9.320576061133271340883240942926

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