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Properties

Label 4-89100-1.1-c1e2-0-1
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 − 3·5-s + 7·11-s + 16-s − 2·19-s − 3·20-s + 4·25-s + 9·29-s − 5·31-s − 3·41-s + 7·44-s + 11·49-s − 21·55-s + 3·59-s + 7·61-s + 64-s − 15·71-s − 2·76-s + 4·79-s − 3·80-s − 3·89-s + 6·95-s + 4·100-s + 18·101-s + 7·109-s + 9·116-s + 32·121-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.34·5-s + 2.11·11-s + 1/4·16-s − 0.458·19-s − 0.670·20-s + 4/5·25-s + 1.67·29-s − 0.898·31-s − 0.468·41-s + 1.05·44-s + 11/7·49-s − 2.83·55-s + 0.390·59-s + 0.896·61-s + 1/8·64-s − 1.78·71-s − 0.229·76-s + 0.450·79-s − 0.335·80-s − 0.317·89-s + 0.615·95-s + 2/5·100-s + 1.79·101-s + 0.670·109-s + 0.835·116-s + 2.90·121-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

\(L(1)\) \(\approx\) \(1.470750350\)
\(L(\frac12)\) \(\approx\) \(1.470750350\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 3 T + p T^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 6 T + p T^{2} ) \)
good7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.a_al
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.17.a_q
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.23.a_ai
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) 2.29.aj_cg
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.f_m
37$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.37.a_bx
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.d_bc
43$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.43.a_ax
47$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \) 2.47.a_cy
53$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \) 2.53.a_br
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.ad_k
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.ah_ek
67$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.67.a_adc
71$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.p_gw
73$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.73.a_acw
79$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.ae_s
83$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.83.a_ax
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.d_eu
97$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \) 2.97.a_acb
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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

−9.635578557165583401757540104558, −8.890367834657714485110302373080, −8.796638991275785730132306916160, −8.185221213300622104917434431314, −7.61467847677440805086191252397, −6.99542043911647788237802968355, −6.81794689223542861837956794795, −6.18783930714934260293815688768, −5.60675952931217470099799770471, −4.62855731301248840345201209580, −4.22355275755443690420076461711, −3.68231195179905413351169814317, −3.11947383474762367047036115421, −2.04121405587109672546314658068, −0.976748933412502195058666591725, 0.976748933412502195058666591725, 2.04121405587109672546314658068, 3.11947383474762367047036115421, 3.68231195179905413351169814317, 4.22355275755443690420076461711, 4.62855731301248840345201209580, 5.60675952931217470099799770471, 6.18783930714934260293815688768, 6.81794689223542861837956794795, 6.99542043911647788237802968355, 7.61467847677440805086191252397, 8.185221213300622104917434431314, 8.796638991275785730132306916160, 8.890367834657714485110302373080, 9.635578557165583401757540104558

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