Properties

Label 4-112e2-1.1-c1e2-0-8
Degree $4$
Conductor $12544$
Sign $-1$
Analytic cond. $0.799816$
Root an. cond. $0.945687$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 8·9-s − 2·17-s − 4·19-s − 8·25-s − 12·27-s − 10·41-s − 7·49-s + 8·51-s + 16·57-s − 20·59-s − 10·73-s + 32·75-s + 23·81-s + 20·83-s + 10·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 6·121-s + 40·123-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 2.30·3-s + 8/3·9-s − 0.485·17-s − 0.917·19-s − 8/5·25-s − 2.30·27-s − 1.56·41-s − 49-s + 1.12·51-s + 2.11·57-s − 2.60·59-s − 1.17·73-s + 3.69·75-s + 23/9·81-s + 2.19·83-s + 1.05·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 6/11·121-s + 3.60·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12544\)    =    \(2^{8} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(0.799816\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{12544} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 12544,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
7$C_2$ \( 1 + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−16.5955433184, −16.3214439874, −15.6216355996, −15.3527979476, −14.8012884938, −13.9704630848, −13.4492092494, −12.9967776131, −12.3793030980, −11.8739578809, −11.6731452885, −11.0387220221, −10.6909788060, −10.1937658629, −9.56981203394, −8.89937800096, −8.03517600042, −7.45189328179, −6.58962985617, −6.22671093794, −5.79315800694, −4.98187403175, −4.55857978785, −3.56274167954, −1.87823786404, 0, 1.87823786404, 3.56274167954, 4.55857978785, 4.98187403175, 5.79315800694, 6.22671093794, 6.58962985617, 7.45189328179, 8.03517600042, 8.89937800096, 9.56981203394, 10.1937658629, 10.6909788060, 11.0387220221, 11.6731452885, 11.8739578809, 12.3793030980, 12.9967776131, 13.4492092494, 13.9704630848, 14.8012884938, 15.3527979476, 15.6216355996, 16.3214439874, 16.5955433184

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