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Properties

Label 4-27200-1.1-c1e2-0-1
Degree $4$
Conductor $27200$
Sign $1$
Analytic cond. $1.73429$
Root an. cond. $1.14757$
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  + 2·5-s − 2·9-s + 8·13-s + 17-s − 25-s − 8·29-s − 4·37-s + 4·41-s − 4·45-s − 2·49-s + 8·53-s + 16·65-s + 8·73-s − 5·81-s + 2·85-s + 20·89-s − 16·97-s − 4·101-s − 16·117-s − 14·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + ⋯
L(s)  = 1  + 0.894·5-s − 2/3·9-s + 2.21·13-s + 0.242·17-s − 1/5·25-s − 1.48·29-s − 0.657·37-s + 0.624·41-s − 0.596·45-s − 2/7·49-s + 1.09·53-s + 1.98·65-s + 0.936·73-s − 5/9·81-s + 0.216·85-s + 2.11·89-s − 1.62·97-s − 0.398·101-s − 1.47·117-s − 1.27·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27200\)    =    \(2^{6} \cdot 5^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(1.73429\)
Root analytic conductor: \(1.14757\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 27200,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.445749461\)
\(L(\frac12)\) \(\approx\) \(1.445749461\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.13.ai_bm
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.19.a_g
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.i_cs
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.31.a_abi
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ae_cs
43$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.43.a_ba
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.53.ai_eo
59$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.59.a_g
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.67.a_acc
71$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.71.a_adu
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.73.ai_gc
79$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.79.a_ck
83$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.83.a_cw
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.89.au_ig
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.q_gc
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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

−10.73223929331675278697278551387, −10.09378239488342768147824104050, −9.438349295489059351740843261312, −9.068020306188082567252754744549, −8.518475361525833097219323694149, −8.069718241044167556804664008263, −7.33421816527718601877474068880, −6.58875079538779119118833166879, −5.96411070273622221177702492634, −5.75195960041086730848401187689, −5.05872792717725842491077125069, −3.90764201231259179046581962329, −3.51486242989554127698315950805, −2.41324139277544213394739581240, −1.41635975867690803296730077237, 1.41635975867690803296730077237, 2.41324139277544213394739581240, 3.51486242989554127698315950805, 3.90764201231259179046581962329, 5.05872792717725842491077125069, 5.75195960041086730848401187689, 5.96411070273622221177702492634, 6.58875079538779119118833166879, 7.33421816527718601877474068880, 8.069718241044167556804664008263, 8.518475361525833097219323694149, 9.068020306188082567252754744549, 9.438349295489059351740843261312, 10.09378239488342768147824104050, 10.73223929331675278697278551387

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