Properties

Degree 4
Conductor $ 7^{2} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 7·3-s + 8·4-s − 7·5-s + 14·6-s + 28·7-s − 40·8-s + 27·9-s + 14·10-s + 5·11-s − 56·12-s − 28·13-s − 56·14-s + 49·15-s + 80·16-s + 21·17-s − 54·18-s − 49·19-s − 56·20-s − 196·21-s − 10·22-s + 159·23-s + 280·24-s + 125·25-s + 56·26-s − 224·27-s + 224·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.34·3-s + 4-s − 0.626·5-s + 0.952·6-s + 1.51·7-s − 1.76·8-s + 9-s + 0.442·10-s + 0.137·11-s − 1.34·12-s − 0.597·13-s − 1.06·14-s + 0.843·15-s + 5/4·16-s + 0.299·17-s − 0.707·18-s − 0.591·19-s − 0.626·20-s − 2.03·21-s − 0.0969·22-s + 1.44·23-s + 2.38·24-s + 25-s + 0.422·26-s − 1.59·27-s + 1.51·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(49\)    =    \(7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{7} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 49,\ (\ :3/2, 3/2),\ 1)$
$L(2)$  $\approx$  $0.357271$
$L(\frac12)$  $\approx$  $0.357271$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 7$, \(F_p\) is a polynomial of degree 4. If $p = 7$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad7$C_2$ \( 1 - 4 p T + p^{3} T^{2} \)
good2$C_2^2$ \( 1 + p T - p^{2} T^{2} + p^{4} T^{3} + p^{6} T^{4} \)
3$C_2^2$ \( 1 + 7 T + 22 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 7 T - 76 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 5 T - 1306 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 14 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 21 T - 4472 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 49 T - 4458 T^{2} + 49 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 159 T + 13114 T^{2} - 159 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 147 T - 8182 T^{2} + 147 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 219 T - 2692 T^{2} + 219 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 350 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 124 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 525 T + 171802 T^{2} + 525 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 303 T - 57068 T^{2} + 303 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 105 T - 194354 T^{2} - 105 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 413 T - 56412 T^{2} - 413 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 415 T - 128538 T^{2} + 415 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 432 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 1113 T + 849752 T^{2} - 1113 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 103 T - 482430 T^{2} - 103 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 1092 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 329 T - 596728 T^{2} - 329 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 882 T + p^{3} T^{2} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−22.75658303906369405879076936982, −21.73262490716289823117921584584, −20.91822196939805874340971609810, −20.83702434468867677778925371131, −19.50153937985057617696689143507, −18.87238470644861754687131275082, −17.75884298513280071891956407611, −17.71699999773205043957589791903, −16.77242482888655925605925920086, −16.09259512717138983717111855239, −14.95994334252734381646800591924, −14.75073122826914210166577738781, −12.63773290481159472105769105934, −11.91232141937324149059092308649, −11.23114614585979948876648558038, −10.79042177854964714767970907634, −9.170771798623879815197307093822, −7.86320333232698952994279195127, −6.69000228021965412375783704525, −5.20239540052751869909136213505, 5.20239540052751869909136213505, 6.69000228021965412375783704525, 7.86320333232698952994279195127, 9.170771798623879815197307093822, 10.79042177854964714767970907634, 11.23114614585979948876648558038, 11.91232141937324149059092308649, 12.63773290481159472105769105934, 14.75073122826914210166577738781, 14.95994334252734381646800591924, 16.09259512717138983717111855239, 16.77242482888655925605925920086, 17.71699999773205043957589791903, 17.75884298513280071891956407611, 18.87238470644861754687131275082, 19.50153937985057617696689143507, 20.83702434468867677778925371131, 20.91822196939805874340971609810, 21.73262490716289823117921584584, 22.75658303906369405879076936982

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