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Properties

Label 4-9600e2-1.1-c1e2-0-3
Degree $4$
Conductor $92160000$
Sign $1$
Analytic cond. $5876.20$
Root an. cond. $8.75536$
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  − 2·3-s + 2·7-s + 3·9-s − 2·13-s + 4·17-s − 2·19-s − 4·21-s − 4·27-s − 4·29-s + 2·31-s + 8·37-s + 4·39-s − 6·43-s + 12·47-s − 5·49-s − 8·51-s + 8·53-s + 4·57-s − 12·59-s − 6·61-s + 6·63-s − 2·67-s − 4·71-s + 16·73-s + 5·81-s − 12·83-s + 8·87-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.755·7-s + 9-s − 0.554·13-s + 0.970·17-s − 0.458·19-s − 0.872·21-s − 0.769·27-s − 0.742·29-s + 0.359·31-s + 1.31·37-s + 0.640·39-s − 0.914·43-s + 1.75·47-s − 5/7·49-s − 1.12·51-s + 1.09·53-s + 0.529·57-s − 1.56·59-s − 0.768·61-s + 0.755·63-s − 0.244·67-s − 0.474·71-s + 1.87·73-s + 5/9·81-s − 1.31·83-s + 0.857·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(92160000\)    =    \(2^{14} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(5876.20\)
Root analytic conductor: \(8.75536\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 92160000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.773287071\)
\(L(\frac12)\) \(\approx\) \(1.773287071\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
good7$D_{4}$ \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_j
11$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.11.a_q
13$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_d
17$D_{4}$ \( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_bg
19$D_{4}$ \( 1 + 2 T + 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_bh
23$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.23.a_bo
29$D_{4}$ \( 1 + 4 T + 56 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_ce
31$D_{4}$ \( 1 - 2 T + 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_cf
37$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_co
41$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.41.a_bc
43$D_{4}$ \( 1 + 6 T + 89 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.43.g_dl
47$D_{4}$ \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_ec
53$D_{4}$ \( 1 - 8 T + 116 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_em
59$D_{4}$ \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_fa
61$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.61.g_fb
67$D_{4}$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_ap
71$D_{4}$ \( 1 + 4 T + 92 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_do
73$D_{4}$ \( 1 - 16 T + 186 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.73.aq_he
79$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.79.a_ck
83$D_{4}$ \( 1 + 12 T + 178 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_gw
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.89.ay_mk
97$D_{4}$ \( 1 - 10 T + 123 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_et
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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

−7.79175911513744194856567433466, −7.53788773815376398377557753054, −7.09686092988116753391478034418, −6.92314589712121599317871590691, −6.30095135380650623051172528914, −6.23717500129851266739051250444, −5.67623283155346112281268606497, −5.58996174526922458848179230024, −5.03025024209131611843374686292, −4.90302197353183404190263933715, −4.44939302289499830631852947975, −4.13680565600817263418219501224, −3.75217654810811447049096624365, −3.29180248913428306873898615497, −2.69937703733811232375044773448, −2.43312412524485642793980704006, −1.70444000065770477889005119259, −1.53535284252747186738376395510, −0.854459975262743152390556109009, −0.40415105857220191814366755280, 0.40415105857220191814366755280, 0.854459975262743152390556109009, 1.53535284252747186738376395510, 1.70444000065770477889005119259, 2.43312412524485642793980704006, 2.69937703733811232375044773448, 3.29180248913428306873898615497, 3.75217654810811447049096624365, 4.13680565600817263418219501224, 4.44939302289499830631852947975, 4.90302197353183404190263933715, 5.03025024209131611843374686292, 5.58996174526922458848179230024, 5.67623283155346112281268606497, 6.23717500129851266739051250444, 6.30095135380650623051172528914, 6.92314589712121599317871590691, 7.09686092988116753391478034418, 7.53788773815376398377557753054, 7.79175911513744194856567433466

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