Properties

Label 4-980000-1.1-c1e2-0-4
Degree $4$
Conductor $980000$
Sign $1$
Analytic cond. $62.4856$
Root an. cond. $2.81154$
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-s + 4-s − 8-s + 3·9-s + 12·13-s + 16-s + 2·17-s − 3·18-s − 12·26-s − 12·29-s − 32-s − 2·34-s + 3·36-s − 16·37-s + 22·41-s + 49-s + 12·52-s − 8·53-s + 12·58-s − 4·61-s + 64-s + 2·68-s − 3·72-s + 14·73-s + 16·74-s − 22·82-s − 22·89-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 9-s + 3.32·13-s + 1/4·16-s + 0.485·17-s − 0.707·18-s − 2.35·26-s − 2.22·29-s − 0.176·32-s − 0.342·34-s + 1/2·36-s − 2.63·37-s + 3.43·41-s + 1/7·49-s + 1.66·52-s − 1.09·53-s + 1.57·58-s − 0.512·61-s + 1/8·64-s + 0.242·68-s − 0.353·72-s + 1.63·73-s + 1.85·74-s − 2.42·82-s − 2.33·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(980000\)    =    \(2^{5} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(62.4856\)
Root analytic conductor: \(2.81154\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{980000} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 980000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.991612160\)
\(L(\frac12)\) \(\approx\) \(1.991612160\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−7.982704647601194275162559588174, −7.929390700558246013782938359361, −7.28202966725007180028492050863, −6.87549347646056013955545596637, −6.43299384802556926626083631868, −5.85064816620516507912571633410, −5.74372284090584252527668413358, −5.11578165943755488210939267812, −4.05801150037857626598528562097, −4.02569848626692601267512154263, −3.49951305711362632263727211152, −2.90932005195183109143742073385, −1.76448136119712918970498775230, −1.58254844871258391875302980534, −0.803015778806315726396220858314, 0.803015778806315726396220858314, 1.58254844871258391875302980534, 1.76448136119712918970498775230, 2.90932005195183109143742073385, 3.49951305711362632263727211152, 4.02569848626692601267512154263, 4.05801150037857626598528562097, 5.11578165943755488210939267812, 5.74372284090584252527668413358, 5.85064816620516507912571633410, 6.43299384802556926626083631868, 6.87549347646056013955545596637, 7.28202966725007180028492050863, 7.929390700558246013782938359361, 7.982704647601194275162559588174

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