Properties

 Degree 2 Conductor $5 \cdot 13 \cdot 31$ Sign $0.707 - 0.707i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + 1.41·3-s − 4-s + i·5-s + 1.00·9-s − 1.41·12-s + (0.707 − 0.707i)13-s + 1.41i·15-s + 16-s + 1.41·17-s + 2i·19-s − i·20-s − 1.41·23-s − 25-s − i·31-s − 1.00·36-s + 1.41i·37-s + ⋯
 L(s)  = 1 + 1.41·3-s − 4-s + i·5-s + 1.00·9-s − 1.41·12-s + (0.707 − 0.707i)13-s + 1.41i·15-s + 16-s + 1.41·17-s + 2i·19-s − i·20-s − 1.41·23-s − 25-s − i·31-s − 1.00·36-s + 1.41i·37-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$2015$$    =    $$5 \cdot 13 \cdot 31$$ $$\varepsilon$$ = $0.707 - 0.707i$ motivic weight = $$0$$ character : $\chi_{2015} (2014, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 2015,\ (\ :0),\ 0.707 - 0.707i)$ $L(\frac{1}{2})$ $\approx$ $1.529637734$ $L(\frac12)$ $\approx$ $1.529637734$ $L(1)$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;13,\;31\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{5,\;13,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 $$1 - iT$$
13 $$1 + (-0.707 + 0.707i)T$$
31 $$1 + iT$$
good2 $$1 + T^{2}$$
3 $$1 - 1.41T + T^{2}$$
7 $$1 + T^{2}$$
11 $$1 + T^{2}$$
17 $$1 - 1.41T + T^{2}$$
19 $$1 - 2iT - T^{2}$$
23 $$1 + 1.41T + T^{2}$$
29 $$1 - T^{2}$$
37 $$1 - 1.41iT - T^{2}$$
41 $$1 - T^{2}$$
43 $$1 - 1.41T + T^{2}$$
47 $$1 + T^{2}$$
53 $$1 - 1.41T + T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - T^{2}$$
67 $$1 + T^{2}$$
71 $$1 + 2iT - T^{2}$$
73 $$1 - 1.41iT - T^{2}$$
79 $$1 - T^{2}$$
83 $$1 + 1.41iT - T^{2}$$
89 $$1 + T^{2}$$
97 $$1 + 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

−9.526048227715514060879179391755, −8.454569638432037948416064631025, −7.905929419800200784613826461383, −7.67248619175886183864425318000, −6.10943406679652768348230402057, −5.62668873291195176499693562043, −4.04180061169350097558750431173, −3.62453946667223181540704568407, −2.89908403831764470521292838867, −1.61494921427731080941020762608, 1.11190893893154543913427395402, 2.38982735964998717767716201328, 3.61139618957688211556248202949, 4.12786225794346189229918619943, 5.04812650819212702495375033823, 5.88231500710522946659600845852, 7.25004716080695604846422553132, 8.021348505173509776035545632566, 8.587365153684021298219123018196, 9.175789832161759726553532010418