Properties

Label 4-112e2-1.1-c1e2-0-14
Degree $4$
Conductor $12544$
Sign $1$
Analytic cond. $0.799816$
Root an. cond. $0.945687$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·3-s + 2·4-s − 4·5-s + 8·6-s + 8·9-s + 8·10-s − 6·11-s − 8·12-s + 16·15-s − 4·16-s − 12·17-s − 16·18-s − 8·19-s − 8·20-s + 12·22-s + 8·25-s − 12·27-s − 2·29-s − 32·30-s + 8·31-s + 8·32-s + 24·33-s + 24·34-s + 16·36-s + 6·37-s + 16·38-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 4-s − 1.78·5-s + 3.26·6-s + 8/3·9-s + 2.52·10-s − 1.80·11-s − 2.30·12-s + 4.13·15-s − 16-s − 2.91·17-s − 3.77·18-s − 1.83·19-s − 1.78·20-s + 2.55·22-s + 8/5·25-s − 2.30·27-s − 0.371·29-s − 5.84·30-s + 1.43·31-s + 1.41·32-s + 4.17·33-s + 4.11·34-s + 8/3·36-s + 0.986·37-s + 2.59·38-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: induced by $\chi_{112} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
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$C_2$ \( 1 + p T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−13.27543726386632237023834736288, −12.77754211622764385398692808552, −11.84585336871961339655873918622, −11.53430190791402638111985601815, −11.12855939747455399766224879908, −11.03615526563313727971497697102, −10.20206174900752305733710802634, −10.17545173493160079326317871982, −8.894601843176605624801842086876, −8.306840649573648139154730327668, −8.126756490343487190475146306068, −7.20565240330284028488283304296, −6.73005728304274755582986553004, −6.31546946995081892233015420942, −5.27249457573811292648996628312, −4.48975487947431727450468077844, −4.32853888064598947732097655777, −2.37699726942469351811773954673, 0, 0, 2.37699726942469351811773954673, 4.32853888064598947732097655777, 4.48975487947431727450468077844, 5.27249457573811292648996628312, 6.31546946995081892233015420942, 6.73005728304274755582986553004, 7.20565240330284028488283304296, 8.126756490343487190475146306068, 8.306840649573648139154730327668, 8.894601843176605624801842086876, 10.17545173493160079326317871982, 10.20206174900752305733710802634, 11.03615526563313727971497697102, 11.12855939747455399766224879908, 11.53430190791402638111985601815, 11.84585336871961339655873918622, 12.77754211622764385398692808552, 13.27543726386632237023834736288

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