Properties

Degree $4$
Conductor $9144576$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 4·7-s + 3·11-s − 13-s + 3·17-s + 5·19-s − 23-s + 5·25-s + 9·29-s − 8·31-s − 4·35-s − 5·37-s + 7·41-s + 3·43-s + 16·47-s + 9·49-s + 9·53-s + 3·55-s − 8·59-s + 4·61-s − 65-s − 24·67-s + 16·71-s + 13·73-s − 12·77-s − 16·79-s + 13·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s + 0.904·11-s − 0.277·13-s + 0.727·17-s + 1.14·19-s − 0.208·23-s + 25-s + 1.67·29-s − 1.43·31-s − 0.676·35-s − 0.821·37-s + 1.09·41-s + 0.457·43-s + 2.33·47-s + 9/7·49-s + 1.23·53-s + 0.404·55-s − 1.04·59-s + 0.512·61-s − 0.124·65-s − 2.93·67-s + 1.89·71-s + 1.52·73-s − 1.36·77-s − 1.80·79-s + 1.42·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3024} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.919071683\)
\(L(\frac12)\) \(\approx\) \(2.919071683\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 7 T + 8 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 3 T - 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 13 T + 86 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
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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

−8.985913729479718967831380398549, −8.732933934379693464505594422958, −8.233174962578292023469683553017, −7.57119411049803624747703675143, −7.33180214280573488193473806053, −7.11633652885132385851304770990, −6.59503516280162287222229234414, −6.29528527917720156835495088659, −5.85777717362417224918877599729, −5.61272125178578237686975088797, −5.14614998307676859804294781221, −4.67411412605140816136168538629, −3.98919237570411684373141638999, −3.87324837390155450402425278720, −3.11240433503660135053255715929, −3.03455879373448703715894101996, −2.44675504335956544698178453999, −1.80732380135353407547154501953, −1.01104755304490151672534901495, −0.65696023828015177518718461867, 0.65696023828015177518718461867, 1.01104755304490151672534901495, 1.80732380135353407547154501953, 2.44675504335956544698178453999, 3.03455879373448703715894101996, 3.11240433503660135053255715929, 3.87324837390155450402425278720, 3.98919237570411684373141638999, 4.67411412605140816136168538629, 5.14614998307676859804294781221, 5.61272125178578237686975088797, 5.85777717362417224918877599729, 6.29528527917720156835495088659, 6.59503516280162287222229234414, 7.11633652885132385851304770990, 7.33180214280573488193473806053, 7.57119411049803624747703675143, 8.233174962578292023469683553017, 8.732933934379693464505594422958, 8.985913729479718967831380398549

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