Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.265 - 0.964i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)4-s + (0.965 + 0.258i)7-s + (−1.22 + 1.22i)13-s + (0.499 − 0.866i)16-s + (0.866 + 0.5i)19-s + (−0.965 + 0.258i)28-s + (1 + 1.73i)31-s + (−1.67 + 0.448i)37-s + (0.866 + 0.499i)49-s + (0.448 − 1.67i)52-s + (0.5 − 0.866i)61-s + 0.999i·64-s + (−0.448 + 1.67i)67-s + (1.67 + 0.448i)73-s − 0.999·76-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)4-s + (0.965 + 0.258i)7-s + (−1.22 + 1.22i)13-s + (0.499 − 0.866i)16-s + (0.866 + 0.5i)19-s + (−0.965 + 0.258i)28-s + (1 + 1.73i)31-s + (−1.67 + 0.448i)37-s + (0.866 + 0.499i)49-s + (0.448 − 1.67i)52-s + (0.5 − 0.866i)61-s + 0.999i·64-s + (−0.448 + 1.67i)67-s + (1.67 + 0.448i)73-s − 0.999·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.265 - 0.964i$
motivic weight  =  \(0\)
character  :  $\chi_{1575} (982, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1575,\ (\ :0),\ 0.265 - 0.964i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.9074108363\)
\(L(\frac12)\)  \(\approx\)  \(0.9074108363\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.965 - 0.258i)T \)
good2 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
17 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
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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

−9.692587009323979735948904014793, −8.880466476375911764988416216830, −8.323083893971741877060416380419, −7.47200143455085660564195714253, −6.76779996749419375197016127954, −5.30369713905360834832976462402, −4.88851783700370456850664719179, −4.01210138426191986275298638833, −2.88198707298615110866155074521, −1.58738803157816210270948662783, 0.792180116850680924500338586414, 2.26155972493654590900433537393, 3.57228677658415322600917372568, 4.71545615224952817906642451696, 5.13033764395005068199996947986, 5.96907758856884649309560275428, 7.29208543049897264809966394489, 7.87690674739160364002794430505, 8.647571403851892205163333089786, 9.558754473303653445368903168761

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