Properties

Degree 2
Conductor $ 2^{6} \cdot 7^{2} $
Sign $-i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + 5-s + i·11-s + i·15-s − 17-s + i·19-s i·23-s + i·27-s + i·31-s − 33-s + 37-s i·47-s i·51-s + 53-s + i·55-s + ⋯
L(s)  = 1  + i·3-s + 5-s + i·11-s + i·15-s − 17-s + i·19-s i·23-s + i·27-s + i·31-s − 33-s + 37-s i·47-s i·51-s + 53-s + i·55-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3136\)    =    \(2^{6} \cdot 7^{2}\)
\( \varepsilon \)  =  $-i$
motivic weight  =  \(0\)
character  :  $\chi_{3136} (1471, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3136,\ (\ :0),\ -i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.485652537\)
\(L(\frac12)\)  \(\approx\)  \(1.485652537\)
\(L(1)\)   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 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 \( 1 - iT - T^{2} \)
5 \( 1 - T + T^{2} \)
11 \( 1 - iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - iT - T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 - iT - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.241876990675967998884559216376, −8.562972428540158871598491415602, −7.52172645344500609831401430436, −6.69096785298671250857312614838, −5.99066298073151943228890797776, −5.05885138415428271470975104212, −4.49012310413206025673875988777, −3.70625648228680756649576216384, −2.48117009337847055691089189396, −1.65953217051652081927372094855, 0.940074708496143179824995466120, 2.02631818263215277663268418528, 2.70051482340089519087629051896, 3.94683850696249150388724611620, 4.98204821140025698768923604444, 5.96847750867158648624665783734, 6.30135256160612369572814490705, 7.16171680186417372324611151230, 7.83043602549637263353919826910, 8.672873226325139418633983288125

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