Properties

Degree 4
Conductor $ 2^{2} \cdot 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 + 5·3-s + 9·5-s − 10·6-s − 28·7-s + 8·8-s + 27·9-s − 18·10-s + 57·11-s − 140·13-s + 56·14-s + 45·15-s − 16·16-s − 51·17-s − 54·18-s − 5·19-s − 140·21-s − 114·22-s − 69·23-s + 40·24-s + 125·25-s + 280·26-s + 280·27-s + 228·29-s − 90·30-s − 23·31-s + 285·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.962·3-s + 0.804·5-s − 0.680·6-s − 1.51·7-s + 0.353·8-s + 9-s − 0.569·10-s + 1.56·11-s − 2.98·13-s + 1.06·14-s + 0.774·15-s − 1/4·16-s − 0.727·17-s − 0.707·18-s − 0.0603·19-s − 1.45·21-s − 1.10·22-s − 0.625·23-s + 0.340·24-s + 25-s + 2.11·26-s + 1.99·27-s + 1.45·29-s − 0.547·30-s − 0.133·31-s + 1.50·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(196\)    =    \(2^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{14} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 196,\ (\ :3/2, 3/2),\ 1)$
$L(2)$  $\approx$  $0.842711$
$L(\frac12)$  $\approx$  $0.842711$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + p T + p^{2} T^{2} \)
7$C_2$ \( 1 + 4 p T + p^{3} T^{2} \)
good3$C_2^2$ \( 1 - 5 T - 2 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 - 9 T - 44 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 57 T + 1918 T^{2} - 57 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 70 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 3 p T - 8 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 5 T - 6834 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 3 p T - 14 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 114 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 23 T - 29262 T^{2} + 23 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 253 T + 13356 T^{2} - 253 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 42 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 124 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 201 T - 63422 T^{2} + 201 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 393 T + 5572 T^{2} - 393 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 219 T - 157418 T^{2} + 219 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 709 T + 275700 T^{2} - 709 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 419 T - 125202 T^{2} + 419 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 96 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 313 T - 291048 T^{2} - 313 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 461 T - 280518 T^{2} + 461 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 588 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 1017 T + 329320 T^{2} - 1017 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 1834 T + p^{3} T^{2} )^{2} \)
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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

−19.70060149722936634821691663140, −19.18029280225304998422878326827, −18.14594227112232935887102024855, −17.62199509974744388632710877392, −16.69027607437785497004687294467, −16.60088533217385288874447020367, −15.43417976203765044553228414374, −14.54147840467397320909262650282, −14.26816558156304400322235008345, −13.29749318027771018416356786078, −12.56656183963219221409419477896, −11.98950982979369141487438511313, −10.17866315908668252753946338987, −9.850467289346286709120806720003, −9.368781952174122019062954225025, −8.545662963510313431303726219308, −7.08174257950588709211683065132, −6.62525929767708568596055772994, −4.56837402176893666395382714770, −2.65328915382901394963039405638, 2.65328915382901394963039405638, 4.56837402176893666395382714770, 6.62525929767708568596055772994, 7.08174257950588709211683065132, 8.545662963510313431303726219308, 9.368781952174122019062954225025, 9.850467289346286709120806720003, 10.17866315908668252753946338987, 11.98950982979369141487438511313, 12.56656183963219221409419477896, 13.29749318027771018416356786078, 14.26816558156304400322235008345, 14.54147840467397320909262650282, 15.43417976203765044553228414374, 16.60088533217385288874447020367, 16.69027607437785497004687294467, 17.62199509974744388632710877392, 18.14594227112232935887102024855, 19.18029280225304998422878326827, 19.70060149722936634821691663140

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