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  :  $\chi_{1575} (457, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1575,\ (\ :0),\ 0.392 - 0.919i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.410098365\)
\(L(\frac12)\)  \(\approx\)  \(1.410098365\)
\(L(1)\)   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

−9.189164220187618661665505634636, −8.583226305235080019589441328021, −8.296591136709525968916893599602, −7.47487465714664736637391474876, −6.45369490331203329685257398117, −6.21907595265765632459265590765, −5.03956849871514225659937931066, −3.77266632992031315665667382987, −2.59423705567609324146549979766, −1.71710089531194582892474345443, 1.31296636736101174825138448459, 2.40365667404626328749019836380, 3.35988578309785708866730364071, 4.14272414368454757616696711833, 5.02183334435202348134933835355, 6.25812986912009067542057710429, 7.21595901499485996283343328513, 8.250527551430187550985831276307, 8.756896816610440773430961172423, 9.674533377327000463156271679046

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