Properties

Label 4-336e2-1.1-c1e2-0-31
Degree $4$
Conductor $112896$
Sign $1$
Analytic cond. $7.19834$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·5-s + 7-s + 3·11-s − 3·15-s + 6·17-s + 2·19-s − 21-s + 25-s + 27-s + 18·29-s − 5·31-s − 3·33-s + 3·35-s − 10·37-s − 12·47-s − 6·49-s − 6·51-s + 9·53-s + 9·55-s − 2·57-s + 9·59-s − 24·67-s − 12·73-s − 75-s + 3·77-s + 9·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s + 0.377·7-s + 0.904·11-s − 0.774·15-s + 1.45·17-s + 0.458·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s + 3.34·29-s − 0.898·31-s − 0.522·33-s + 0.507·35-s − 1.64·37-s − 1.75·47-s − 6/7·49-s − 0.840·51-s + 1.23·53-s + 1.21·55-s − 0.264·57-s + 1.17·59-s − 2.93·67-s − 1.40·73-s − 0.115·75-s + 0.341·77-s + 1.01·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(112896\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(7.19834\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{336} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 112896,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.917857798\)
\(L(\frac12)\) \(\approx\) \(1.917857798\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 24 T + 259 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T + 101 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−11.86955287888147306674300484446, −11.44745381110301718548587896288, −10.63515159240329913830380747356, −10.46377590517617138320775756959, −9.819315281793336142955598128581, −9.776979601332275083691057184293, −9.031441912788082727672425083054, −8.531191304456363468334837613519, −8.138714065745209770547273243674, −7.42819210345826005041097765573, −6.65753218298887524195105826938, −6.62133921340766585861987217264, −5.68545939867883456956698803485, −5.63013305515793625995455845003, −4.91742076453252468906770468423, −4.38546512750434789043718867075, −3.41515829207316476287489313638, −2.86885285692459854617263203460, −1.76568093759239594602061038249, −1.19226971877493348484139312883, 1.19226971877493348484139312883, 1.76568093759239594602061038249, 2.86885285692459854617263203460, 3.41515829207316476287489313638, 4.38546512750434789043718867075, 4.91742076453252468906770468423, 5.63013305515793625995455845003, 5.68545939867883456956698803485, 6.62133921340766585861987217264, 6.65753218298887524195105826938, 7.42819210345826005041097765573, 8.138714065745209770547273243674, 8.531191304456363468334837613519, 9.031441912788082727672425083054, 9.776979601332275083691057184293, 9.819315281793336142955598128581, 10.46377590517617138320775756959, 10.63515159240329913830380747356, 11.44745381110301718548587896288, 11.86955287888147306674300484446

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