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Properties

Label 4-89100-1.1-c1e2-0-2
Degree $4$
Conductor $89100$
Sign $1$
Analytic cond. $5.68109$
Root an. cond. $1.54386$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 2·5-s − 3·11-s + 16-s + 8·19-s + 2·20-s − 25-s + 4·29-s + 12·41-s − 3·44-s − 14·49-s − 6·55-s + 8·59-s + 12·61-s + 64-s + 8·76-s − 16·79-s + 2·80-s + 12·89-s + 16·95-s − 100-s − 12·101-s + 12·109-s + 4·116-s + 2·121-s − 12·125-s + 127-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.894·5-s − 0.904·11-s + 1/4·16-s + 1.83·19-s + 0.447·20-s − 1/5·25-s + 0.742·29-s + 1.87·41-s − 0.452·44-s − 2·49-s − 0.809·55-s + 1.04·59-s + 1.53·61-s + 1/8·64-s + 0.917·76-s − 1.80·79-s + 0.223·80-s + 1.27·89-s + 1.64·95-s − 0.0999·100-s − 1.19·101-s + 1.14·109-s + 0.371·116-s + 2/11·121-s − 1.07·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(89100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(5.68109\)
Root analytic conductor: \(1.54386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 89100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.994142562\)
\(L(\frac12)\) \(\approx\) \(1.994142562\)
\(L(\frac{3}{2})\) 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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.17.a_abi
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.ae_ac
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.a_acg
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.47.a_abi
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.53.a_bm
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.ai_cs
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.am_dq
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.q_gc
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.97.a_ck
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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

−9.610220250112460692493308298453, −9.430232981659450717816536191731, −8.640688281335016483995815989544, −8.094133610796075829652777105742, −7.62721206604976680056662914204, −7.19822932758320151585445727874, −6.54769610605873137970416123162, −6.03217126969705284384505432749, −5.45069360999926371989222959779, −5.16528826429793492681073482786, −4.35348532709281903574871508763, −3.44137187501834711872066697760, −2.79927633660285240504213825896, −2.20082795018380216278956532106, −1.16286281575520808065038381791, 1.16286281575520808065038381791, 2.20082795018380216278956532106, 2.79927633660285240504213825896, 3.44137187501834711872066697760, 4.35348532709281903574871508763, 5.16528826429793492681073482786, 5.45069360999926371989222959779, 6.03217126969705284384505432749, 6.54769610605873137970416123162, 7.19822932758320151585445727874, 7.62721206604976680056662914204, 8.094133610796075829652777105742, 8.640688281335016483995815989544, 9.430232981659450717816536191731, 9.610220250112460692493308298453

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