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Properties

Label 4-1520e2-1.1-c1e2-0-20
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 − 2·15-s + 7·19-s + 15·23-s + 10·27-s + 6·29-s + 8·31-s − 15·41-s − 3·45-s + 6·47-s + 11·49-s + 9·53-s + 14·57-s − 12·59-s − 10·61-s − 4·67-s + 30·69-s − 6·71-s + 2·73-s + 4·79-s + 20·81-s + 12·87-s − 27·89-s + 16·93-s − 7·95-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 9-s − 0.516·15-s + 1.60·19-s + 3.12·23-s + 1.92·27-s + 1.11·29-s + 1.43·31-s − 2.34·41-s − 0.447·45-s + 0.875·47-s + 11/7·49-s + 1.23·53-s + 1.85·57-s − 1.56·59-s − 1.28·61-s − 0.488·67-s + 3.61·69-s − 0.712·71-s + 0.234·73-s + 0.450·79-s + 20/9·81-s + 1.28·87-s − 2.86·89-s + 1.65·93-s − 0.718·95-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

\(L(1)\) \(\approx\) \(4.473133043\)
\(L(\frac12)\) \(\approx\) \(4.473133043\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 7 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_b
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.a_al
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.11.a_f
13$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.13.a_n
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
23$C_2^2$ \( 1 - 15 T + 98 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.23.ap_du
29$C_2^2$ \( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_bp
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2^2$ \( 1 - 71 T^{2} + p^{2} T^{4} \) 2.37.a_act
41$C_2^2$ \( 1 + 15 T + 116 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.41.p_em
43$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.43.a_br
47$C_2^2$ \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_ch
53$C_2^2$ \( 1 - 9 T + 80 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.53.aj_dc
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.61.k_bn
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_abz
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_abj
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_acr
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.ae_acl
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.83.a_acg
89$C_2^2$ \( 1 + 27 T + 332 T^{2} + 27 p T^{3} + p^{2} T^{4} \) 2.89.bb_mu
97$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.97.a_dt
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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

−9.447626019423920204106581577311, −9.269219945017920319721471282599, −8.731150363683768213043298923453, −8.416679428359766017071750142211, −8.413349294140568456685132473102, −7.49977610956217201122354973484, −7.30043268635282368455226344325, −7.10497220494914941409037051427, −6.59176000995694541688894858067, −6.11002843020216393465684932479, −5.33531315168600516087188028764, −5.06345955298462700376778508711, −4.52138550344180470020338037542, −4.31337341116496284426662432702, −3.33857768886775966916876679723, −3.02974719571488939392396967675, −3.02411444417152151679190006529, −2.21040810276674802692160082177, −1.14461240640648834349234863454, −0.999980426891274252779301264080, 0.999980426891274252779301264080, 1.14461240640648834349234863454, 2.21040810276674802692160082177, 3.02411444417152151679190006529, 3.02974719571488939392396967675, 3.33857768886775966916876679723, 4.31337341116496284426662432702, 4.52138550344180470020338037542, 5.06345955298462700376778508711, 5.33531315168600516087188028764, 6.11002843020216393465684932479, 6.59176000995694541688894858067, 7.10497220494914941409037051427, 7.30043268635282368455226344325, 7.49977610956217201122354973484, 8.413349294140568456685132473102, 8.416679428359766017071750142211, 8.731150363683768213043298923453, 9.269219945017920319721471282599, 9.447626019423920204106581577311

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