Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $-0.745 + 0.666i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.260 + 0.260i)2-s + (−0.826 − 1.52i)3-s + 1.86i·4-s + (0.611 + 0.180i)6-s + (−0.707 − 0.707i)7-s + (−1.00 − 1.00i)8-s + (−1.63 + 2.51i)9-s − 3.38i·11-s + (2.83 − 1.54i)12-s + (−1.59 + 1.59i)13-s + 0.368·14-s − 3.20·16-s + (−0.140 + 0.140i)17-s + (−0.230 − 1.07i)18-s − 7.34i·19-s + ⋯
L(s)  = 1  + (−0.184 + 0.184i)2-s + (−0.477 − 0.878i)3-s + 0.932i·4-s + (0.249 + 0.0738i)6-s + (−0.267 − 0.267i)7-s + (−0.355 − 0.355i)8-s + (−0.544 + 0.838i)9-s − 1.02i·11-s + (0.819 − 0.445i)12-s + (−0.442 + 0.442i)13-s + 0.0983·14-s − 0.801·16-s + (−0.0341 + 0.0341i)17-s + (−0.0542 − 0.254i)18-s − 1.68i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.745 + 0.666i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (407, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ -0.745 + 0.666i)\)
\(L(1)\)  \(\approx\)  \(0.144657 - 0.378828i\)
\(L(\frac12)\)  \(\approx\)  \(0.144657 - 0.378828i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.826 + 1.52i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (0.260 - 0.260i)T - 2iT^{2} \)
11 \( 1 + 3.38iT - 11T^{2} \)
13 \( 1 + (1.59 - 1.59i)T - 13iT^{2} \)
17 \( 1 + (0.140 - 0.140i)T - 17iT^{2} \)
19 \( 1 + 7.34iT - 19T^{2} \)
23 \( 1 + (2.21 + 2.21i)T + 23iT^{2} \)
29 \( 1 + 9.49T + 29T^{2} \)
31 \( 1 - 0.922T + 31T^{2} \)
37 \( 1 + (5.91 + 5.91i)T + 37iT^{2} \)
41 \( 1 - 1.39iT - 41T^{2} \)
43 \( 1 + (0.864 - 0.864i)T - 43iT^{2} \)
47 \( 1 + (0.651 - 0.651i)T - 47iT^{2} \)
53 \( 1 + (6.54 + 6.54i)T + 53iT^{2} \)
59 \( 1 + 6.25T + 59T^{2} \)
61 \( 1 - 1.83T + 61T^{2} \)
67 \( 1 + (-0.815 - 0.815i)T + 67iT^{2} \)
71 \( 1 - 9.77iT - 71T^{2} \)
73 \( 1 + (-4.80 + 4.80i)T - 73iT^{2} \)
79 \( 1 + 3.41iT - 79T^{2} \)
83 \( 1 + (-6.26 - 6.26i)T + 83iT^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + (-6.71 - 6.71i)T + 97iT^{2} \)
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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

−10.87106973016819798044159413609, −9.373324438508521638874877942886, −8.608085724301474203601418518561, −7.66563407232733989010420177698, −6.97372078491655678893032713830, −6.19200375930075878637188255902, −4.92552606883980296074893552898, −3.54406884077322847349767370885, −2.32837060391583050993418102030, −0.25339454953567200242648595617, 1.85136750564701038751420504675, 3.50356489099745560070458077581, 4.74457092784149894741012380469, 5.57364272092875417625789733436, 6.31189394448554660798567686282, 7.61593865050526716991310656127, 8.927340702650658543023679985057, 9.797614878456276693142400712621, 10.09711375639760593771939074642, 10.98034814328837303738610040018

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