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Properties

Label 4-3480e2-1.1-c1e2-0-11
Degree $4$
Conductor $12110400$
Sign $1$
Analytic cond. $772.169$
Root an. cond. $5.27142$
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 + 2·5-s + 7-s + 3·9-s + 11-s + 5·13-s + 4·15-s + 3·17-s + 8·19-s + 2·21-s + 8·23-s + 3·25-s + 4·27-s + 2·29-s − 4·31-s + 2·33-s + 2·35-s − 10·37-s + 10·39-s − 8·41-s − 10·43-s + 6·45-s + 7·47-s − 9·49-s + 6·51-s − 6·53-s + 2·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 0.377·7-s + 9-s + 0.301·11-s + 1.38·13-s + 1.03·15-s + 0.727·17-s + 1.83·19-s + 0.436·21-s + 1.66·23-s + 3/5·25-s + 0.769·27-s + 0.371·29-s − 0.718·31-s + 0.348·33-s + 0.338·35-s − 1.64·37-s + 1.60·39-s − 1.24·41-s − 1.52·43-s + 0.894·45-s + 1.02·47-s − 9/7·49-s + 0.840·51-s − 0.824·53-s + 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12110400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12110400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12110400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(772.169\)
Root analytic conductor: \(5.27142\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12110400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.467358499\)
\(L(\frac12)\) \(\approx\) \(8.467358499\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
29$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) 2.7.ab_k
11$D_{4}$ \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_s
13$D_{4}$ \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.13.af_bc
17$D_{4}$ \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_bg
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.23.ai_ck
31$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_ac
37$D_{4}$ \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.37.k_de
41$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_be
43$D_{4}$ \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_dq
47$D_{4}$ \( 1 - 7 T + 102 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.47.ah_dy
53$D_{4}$ \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_du
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_cs
67$D_{4}$ \( 1 - 3 T + 30 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.67.ad_be
71$D_{4}$ \( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.71.ac_ew
73$D_{4}$ \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_fi
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$D_{4}$ \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_fe
89$D_{4}$ \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.89.ah_dg
97$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_bq
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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

−8.720748453953804990023593105745, −8.581510003872410901146493975645, −7.939032316471038849532849341693, −7.907896312069253172322104653193, −7.18688487881422149486291501741, −7.04811121401955831763877638323, −6.51396622013579905175007817657, −6.38610032294331897163552789587, −5.52829861606563717789666439080, −5.42530868124087232264572227755, −4.94973844894410448830168837325, −4.76650737803926638518016847568, −3.81381074792260064834914904458, −3.59248335689119721197743422850, −3.12516177763803609415360403088, −3.10278892780184392393689431027, −2.12613722404525038295746779069, −1.81942565094386554078288588296, −1.23442023144852245251640699097, −0.931683207846374921169772203959, 0.931683207846374921169772203959, 1.23442023144852245251640699097, 1.81942565094386554078288588296, 2.12613722404525038295746779069, 3.10278892780184392393689431027, 3.12516177763803609415360403088, 3.59248335689119721197743422850, 3.81381074792260064834914904458, 4.76650737803926638518016847568, 4.94973844894410448830168837325, 5.42530868124087232264572227755, 5.52829861606563717789666439080, 6.38610032294331897163552789587, 6.51396622013579905175007817657, 7.04811121401955831763877638323, 7.18688487881422149486291501741, 7.907896312069253172322104653193, 7.939032316471038849532849341693, 8.581510003872410901146493975645, 8.720748453953804990023593105745

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