# 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

# Related objects

## 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 : $\chi_{525} (218, \cdot )$ 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

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