Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.392 - 0.919i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.707 − 0.707i)3-s + (0.500 − 0.866i)6-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (−0.965 + 0.258i)13-s + (−0.866 − 0.500i)14-s + (−0.5 + 0.866i)16-s + (0.258 + 0.965i)17-s + (−0.965 + 0.258i)18-s + (−0.866 + 0.5i)19-s + 1.00·21-s − 0.999i·24-s + (−0.499 − 0.866i)26-s + (0.707 − 0.707i)27-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.707 − 0.707i)3-s + (0.500 − 0.866i)6-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (−0.965 + 0.258i)13-s + (−0.866 − 0.500i)14-s + (−0.5 + 0.866i)16-s + (0.258 + 0.965i)17-s + (−0.965 + 0.258i)18-s + (−0.866 + 0.5i)19-s + 1.00·21-s − 0.999i·24-s + (−0.499 − 0.866i)26-s + (0.707 − 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.392 - 0.919i$
motivic weight  =  \(0\)
character  :  $\chi_{1575} (193, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1575,\ (\ :0),\ -0.392 - 0.919i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.9054942953\)
\(L(\frac12)\)  \(\approx\)  \(0.9054942953\)
\(L(1)\)   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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + 2T + T^{2} \)
73 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
89 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
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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

−10.11015407920497134423471998874, −8.713613954801187615977324892021, −8.058327680695988149749650229176, −7.25771320004280098029946632312, −6.32942350231249235149366614891, −6.20943784108573504351208588879, −5.21266842565907864554452232935, −4.43204537611414989455405025752, −2.74422780804718044332332636580, −1.73525798924145876999627885531, 0.70464436383444608646009294590, 2.50623452848331492808740366331, 3.32193752729075552423907321975, 4.28421835543227617220362949836, 4.85139110737766433532986403906, 6.07067229145927456801669299032, 6.93974041419358657816894768104, 7.53172417949998905016521782527, 8.993908124173936070707999165623, 9.749032789423466325775889919518

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