Properties

Label 4-901404-1.1-c1e2-0-1
Degree $4$
Conductor $901404$
Sign $1$
Analytic cond. $57.4743$
Root an. cond. $2.75339$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4-s + 7-s + 9-s − 2·12-s + 16-s + 12·19-s − 2·21-s − 2·25-s + 4·27-s + 28-s − 8·31-s + 36-s − 12·37-s − 20·43-s − 2·48-s + 49-s − 24·57-s + 12·61-s + 63-s + 64-s + 8·67-s − 15·73-s + 4·75-s + 12·76-s + 16·79-s − 11·81-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 0.377·7-s + 1/3·9-s − 0.577·12-s + 1/4·16-s + 2.75·19-s − 0.436·21-s − 2/5·25-s + 0.769·27-s + 0.188·28-s − 1.43·31-s + 1/6·36-s − 1.97·37-s − 3.04·43-s − 0.288·48-s + 1/7·49-s − 3.17·57-s + 1.53·61-s + 0.125·63-s + 1/8·64-s + 0.977·67-s − 1.75·73-s + 0.461·75-s + 1.37·76-s + 1.80·79-s − 1.22·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(901404\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{3} \cdot 73\)
Sign: $1$
Analytic conductor: \(57.4743\)
Root analytic conductor: \(2.75339\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{901404} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 901404,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.376835233\)
\(L(\frac12)\) \(\approx\) \(1.376835233\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
7$C_1$ \( 1 - T \)
73$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 14 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.104896712313343120997320633635, −7.44591300444563061205382192540, −7.38143668393976292143625351940, −6.80000171759750942302082249743, −6.46297831799875791051495100213, −5.83721704756890568930732991673, −5.36981291871743877546857941859, −5.14030387392133989849927045411, −4.90082812907387706804743696889, −3.89423478851217689693985034568, −3.36141597922135942136294390043, −3.09179969440150036582632462395, −1.99354823677226090726908316526, −1.53607576198637436986990905357, −0.60845760105563453073349543843, 0.60845760105563453073349543843, 1.53607576198637436986990905357, 1.99354823677226090726908316526, 3.09179969440150036582632462395, 3.36141597922135942136294390043, 3.89423478851217689693985034568, 4.90082812907387706804743696889, 5.14030387392133989849927045411, 5.36981291871743877546857941859, 5.83721704756890568930732991673, 6.46297831799875791051495100213, 6.80000171759750942302082249743, 7.38143668393976292143625351940, 7.44591300444563061205382192540, 8.104896712313343120997320633635

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