Properties

Label 4-1400e2-1.1-c1e2-0-2
Degree $4$
Conductor $1960000$
Sign $1$
Analytic cond. $124.971$
Root an. cond. $3.34350$
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  − 2·3-s − 4·7-s + 3·9-s + 11-s + 6·13-s − 2·17-s + 5·19-s + 8·21-s + 7·23-s − 10·27-s − 12·29-s − 4·31-s − 2·33-s − 5·37-s − 12·39-s − 10·41-s − 12·43-s − 9·47-s + 9·49-s + 4·51-s + 11·53-s − 10·57-s − 8·59-s + 12·61-s − 12·63-s − 4·67-s − 14·69-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.51·7-s + 9-s + 0.301·11-s + 1.66·13-s − 0.485·17-s + 1.14·19-s + 1.74·21-s + 1.45·23-s − 1.92·27-s − 2.22·29-s − 0.718·31-s − 0.348·33-s − 0.821·37-s − 1.92·39-s − 1.56·41-s − 1.82·43-s − 1.31·47-s + 9/7·49-s + 0.560·51-s + 1.51·53-s − 1.32·57-s − 1.04·59-s + 1.53·61-s − 1.51·63-s − 0.488·67-s − 1.68·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1960000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(124.971\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5429231206\)
\(L(\frac12)\) \(\approx\) \(0.5429231206\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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

−10.08921634039807693777943196528, −9.227195249067869121867676290485, −9.218232201703958353331722765946, −8.686558791093811603843156092851, −8.213883586280911004510475687181, −7.53348643410263489050637756703, −7.16885078576524821138669290795, −6.80617617349166798761665443841, −6.51431378284343142968756851234, −6.11255172395839609847995573404, −5.59701052952267362423131515776, −5.24381242950577318557947562524, −5.04317126478487142612273099994, −3.94112991422782330077792513114, −3.78701812917989871350363925335, −3.42593979867028206654495862237, −2.87048462374737838085946338544, −1.72741785130060712862589600306, −1.43443711147013029053496661736, −0.33464136762397425420927785505, 0.33464136762397425420927785505, 1.43443711147013029053496661736, 1.72741785130060712862589600306, 2.87048462374737838085946338544, 3.42593979867028206654495862237, 3.78701812917989871350363925335, 3.94112991422782330077792513114, 5.04317126478487142612273099994, 5.24381242950577318557947562524, 5.59701052952267362423131515776, 6.11255172395839609847995573404, 6.51431378284343142968756851234, 6.80617617349166798761665443841, 7.16885078576524821138669290795, 7.53348643410263489050637756703, 8.213883586280911004510475687181, 8.686558791093811603843156092851, 9.218232201703958353331722765946, 9.227195249067869121867676290485, 10.08921634039807693777943196528

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