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Properties

Label 4-7168-1.1-c1e2-0-1
Degree $4$
Conductor $7168$
Sign $1$
Analytic cond. $0.457037$
Root an. cond. $0.822220$
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  + 3·7-s − 2·9-s + 2·25-s − 4·31-s − 12·41-s − 12·47-s + 6·49-s − 6·63-s + 12·71-s + 16·73-s − 4·79-s − 5·81-s − 8·97-s − 16·103-s − 12·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯
L(s)  = 1  + 1.13·7-s − 2/3·9-s + 2/5·25-s − 0.718·31-s − 1.87·41-s − 1.75·47-s + 6/7·49-s − 0.755·63-s + 1.42·71-s + 1.87·73-s − 0.450·79-s − 5/9·81-s − 0.812·97-s − 1.57·103-s − 1.12·113-s + 2/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7168\)    =    \(2^{10} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.457037\)
Root analytic conductor: \(0.822220\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7168,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9616664412\)
\(L(\frac12)\) \(\approx\) \(0.9616664412\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.13.a_ac
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.e_be
37$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.37.a_w
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.m_dq
53$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.53.a_w
59$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.59.a_ack
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.61.a_aby
67$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.67.a_acw
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.am_fm
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.73.aq_gs
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.e_ew
83$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.83.a_abm
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.i_gs
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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

−11.69024980118819473856887104243, −11.34445342018361327450231484678, −10.86576189410051471253092922205, −10.22787479361321949304417869979, −9.533045362774965054290771507438, −8.885024905128671255349054244173, −8.202744504966320110129191973462, −7.989371888331470905910455150699, −6.99755185404804050545463733579, −6.45588407167576011914734961027, −5.36787297727723195486330194633, −5.10123263746544124092940502080, −4.07787932148272594662387398721, −3.07451733941196317434654075021, −1.81717030273873633954336446889, 1.81717030273873633954336446889, 3.07451733941196317434654075021, 4.07787932148272594662387398721, 5.10123263746544124092940502080, 5.36787297727723195486330194633, 6.45588407167576011914734961027, 6.99755185404804050545463733579, 7.989371888331470905910455150699, 8.202744504966320110129191973462, 8.885024905128671255349054244173, 9.533045362774965054290771507438, 10.22787479361321949304417869979, 10.86576189410051471253092922205, 11.34445342018361327450231484678, 11.69024980118819473856887104243

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