Properties

Degree 2
Conductor $ 2^{6} \cdot 7^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 9-s − 4·13-s − 6·17-s − 2·19-s − 5·25-s − 4·27-s + 6·29-s − 4·31-s − 2·37-s − 8·39-s − 6·41-s + 8·43-s − 12·47-s − 12·51-s − 6·53-s − 4·57-s + 6·59-s + 8·61-s − 4·67-s − 2·73-s − 10·75-s − 8·79-s − 11·81-s + 6·83-s + 12·87-s + 6·89-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s − 1.10·13-s − 1.45·17-s − 0.458·19-s − 25-s − 0.769·27-s + 1.11·29-s − 0.718·31-s − 0.328·37-s − 1.28·39-s − 0.937·41-s + 1.21·43-s − 1.75·47-s − 1.68·51-s − 0.824·53-s − 0.529·57-s + 0.781·59-s + 1.02·61-s − 0.488·67-s − 0.234·73-s − 1.15·75-s − 0.900·79-s − 1.22·81-s + 0.658·83-s + 1.28·87-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3136\)    =    \(2^{6} \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{3136} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 3136,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.23995681113974, −18.25075551904641, −17.49183126900907, −17.24592561480089, −16.14059429647331, −15.65849029962476, −14.90752478083503, −14.53183250029224, −13.85001645200284, −13.24725603136660, −12.66981392155921, −11.79131588378276, −11.18221295820956, −10.24229190635021, −9.649484617797022, −8.934540413043952, −8.408228795468761, −7.689573897205061, −6.961752653738520, −6.173689243841099, −5.067018248475824, −4.315476230448915, −3.424431844618837, −2.497394578680223, −1.909968310823370, 0, 1.909968310823370, 2.497394578680223, 3.424431844618837, 4.315476230448915, 5.067018248475824, 6.173689243841099, 6.961752653738520, 7.689573897205061, 8.408228795468761, 8.934540413043952, 9.649484617797022, 10.24229190635021, 11.18221295820956, 11.79131588378276, 12.66981392155921, 13.24725603136660, 13.85001645200284, 14.53183250029224, 14.90752478083503, 15.65849029962476, 16.14059429647331, 17.24592561480089, 17.49183126900907, 18.25075551904641, 19.23995681113974

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