Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 5^{2} \cdot 7$ Sign $0.988 - 0.153i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2.54 − 0.717i)7-s − 0.999·8-s + (−5.09 + 2.94i)11-s + 4.05·13-s + (−1.89 + 1.84i)14-s + (−0.5 + 0.866i)16-s + (−0.371 + 0.214i)17-s + (−5.30 − 3.06i)19-s + 5.88i·22-s + (0.876 − 1.51i)23-s + (2.02 − 3.51i)26-s + (0.651 + 2.56i)28-s − 0.0419i·29-s + ⋯
 L(s)  = 1 + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.962 − 0.271i)7-s − 0.353·8-s + (−1.53 + 0.886i)11-s + 1.12·13-s + (−0.506 + 0.493i)14-s + (−0.125 + 0.216i)16-s + (−0.0901 + 0.0520i)17-s + (−1.21 − 0.703i)19-s + 1.25i·22-s + (0.182 − 0.316i)23-s + (0.397 − 0.689i)26-s + (0.123 + 0.484i)28-s − 0.00778i·29-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$3150$$    =    $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $0.988 - 0.153i$ motivic weight = $$1$$ character : $\chi_{3150} (1349, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 3150,\ (\ :1/2),\ 0.988 - 0.153i)$ $L(1)$ $\approx$ $1.296741980$ $L(\frac12)$ $\approx$ $1.296741980$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-0.5 + 0.866i)T$$
3 $$1$$
5 $$1$$
7 $$1 + (2.54 + 0.717i)T$$
good11 $$1 + (5.09 - 2.94i)T + (5.5 - 9.52i)T^{2}$$
13 $$1 - 4.05T + 13T^{2}$$
17 $$1 + (0.371 - 0.214i)T + (8.5 - 14.7i)T^{2}$$
19 $$1 + (5.30 + 3.06i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + (-0.876 + 1.51i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + 0.0419iT - 29T^{2}$$
31 $$1 + (-7.92 + 4.57i)T + (15.5 - 26.8i)T^{2}$$
37 $$1 + (0.928 + 0.536i)T + (18.5 + 32.0i)T^{2}$$
41 $$1 - 8.61T + 41T^{2}$$
43 $$1 - 11.0iT - 43T^{2}$$
47 $$1 + (-0.834 - 0.481i)T + (23.5 + 40.7i)T^{2}$$
53 $$1 + (-6.57 - 11.3i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (-6.77 - 11.7i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-1.05 - 0.609i)T + (30.5 + 52.8i)T^{2}$$
67 $$1 + (10.9 - 6.32i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + 2.54iT - 71T^{2}$$
73 $$1 + (-4.66 - 8.08i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (-5.35 + 9.28i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 - 10.1iT - 83T^{2}$$
89 $$1 + (3.15 - 5.46i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 - 2.59T + 97T^{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

−8.816757437735764745179441127631, −8.004385336213393004200425883964, −7.14016618603056949263483501638, −6.28368346002039486472286499184, −5.71020719134966144993369251148, −4.52654110871430658038560005374, −4.13822229285877941384120349627, −2.83939938883491637181975728627, −2.44518297567055554712447998822, −0.901497623208167433432595597644, 0.44451454640367227495175950257, 2.28821911513956848597004415782, 3.23912962915868544016193281607, 3.85970470192378654486212007881, 5.01968802551127059607314140918, 5.76646916197183777385417733160, 6.27868009570827519500149593279, 7.00193551670167206136011866348, 8.088197937786961028505424315676, 8.436308406439179641438432193232