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Properties

Label 4-12675-1.1-c1e2-0-1
Degree $4$
Conductor $12675$
Sign $1$
Analytic cond. $0.808168$
Root an. cond. $0.948146$
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  + 3-s + 4-s + 12-s + 4·13-s − 3·16-s − 6·23-s − 25-s − 4·27-s + 4·39-s + 10·43-s − 3·48-s − 10·49-s + 4·52-s + 8·61-s − 7·64-s − 6·69-s − 75-s − 4·79-s − 7·81-s − 6·92-s − 100-s − 12·101-s + 22·103-s − 18·107-s − 4·108-s − 14·121-s + 127-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 0.288·12-s + 1.10·13-s − 3/4·16-s − 1.25·23-s − 1/5·25-s − 0.769·27-s + 0.640·39-s + 1.52·43-s − 0.433·48-s − 1.42·49-s + 0.554·52-s + 1.02·61-s − 7/8·64-s − 0.722·69-s − 0.115·75-s − 0.450·79-s − 7/9·81-s − 0.625·92-s − 0.0999·100-s − 1.19·101-s + 2.16·103-s − 1.74·107-s − 0.384·108-s − 1.27·121-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12675\)    =    \(3 \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.808168\)
Root analytic conductor: \(0.948146\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12675,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.351407444\)
\(L(\frac12)\) \(\approx\) \(1.351407444\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
5$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.2.a_ab
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.g_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.a_abm
37$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.37.a_bu
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.41.a_abu
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.43.ak_dy
47$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.47.a_bm
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.59.a_ck
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.ai_dy
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.a_ack
71$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.71.a_o
73$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.73.a_ac
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.e_ew
83$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.83.a_abi
89$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.89.a_ade
97$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.97.a_bi
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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

−11.27716196158581691121235539945, −10.85771379277942351835626714003, −10.18165439846856238700279054637, −9.581157913637604611185571767775, −9.054864374844616629693626015830, −8.458769571359115272271134818300, −7.936524194158162055059725288999, −7.37762195504298255774482026415, −6.57835162906456913510049305841, −6.10616765189151137631829497972, −5.40428411438070219587341002587, −4.31126583371613296422474005817, −3.73773869400948077575147092793, −2.75117203203167271628319449010, −1.81847035648135804639724190950, 1.81847035648135804639724190950, 2.75117203203167271628319449010, 3.73773869400948077575147092793, 4.31126583371613296422474005817, 5.40428411438070219587341002587, 6.10616765189151137631829497972, 6.57835162906456913510049305841, 7.37762195504298255774482026415, 7.936524194158162055059725288999, 8.458769571359115272271134818300, 9.054864374844616629693626015830, 9.581157913637604611185571767775, 10.18165439846856238700279054637, 10.85771379277942351835626714003, 11.27716196158581691121235539945

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