Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.997 - 0.0677i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.997 - 0.0677i$
motivic weight  =  \(0\)
character  :  $\chi_{1575} (1243, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1575,\ (\ :0),\ 0.997 - 0.0677i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.8618631498\)
\(L(\frac12)\)  \(\approx\)  \(0.8618631498\)
\(L(1)\)   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

−9.633245292395528928581285151721, −8.956759657588321196914807279888, −8.292352827898414911085895395407, −7.11309569101728868152936163169, −6.18863595605959618555640744207, −5.73078654181710156075834357063, −4.47505599178658663483744087081, −3.87278746461644785201095187842, −2.65949402407712535131607269334, −1.09775044191987694214685054986, 0.949418929156009399994329756279, 3.05256555670321683265318787858, 3.46911862137714315463332035950, 4.51282921064172630314673716425, 5.55448256645153455875480438231, 6.27851540783955498472099823784, 7.36208024937111332189292436558, 8.115205156186103265466106685588, 8.768755235122828847830087665830, 9.629720401514810864897360105557

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