# Properties

 Degree 2 Conductor $13 \cdot 31$ Sign $-0.913 - 0.407i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.01i·2-s − 1.05·3-s + 0.974·4-s + 3.24i·5-s − 1.06i·6-s + 1.79i·7-s + 3.01i·8-s − 1.89·9-s − 3.28·10-s − 6.09i·11-s − 1.02·12-s + (−1.47 + 3.29i)13-s − 1.81·14-s − 3.41i·15-s − 1.09·16-s + 4.79·17-s + ⋯
 L(s)  = 1 + 0.715i·2-s − 0.607·3-s + 0.487·4-s + 1.45i·5-s − 0.434i·6-s + 0.678i·7-s + 1.06i·8-s − 0.631·9-s − 1.03·10-s − 1.83i·11-s − 0.295·12-s + (−0.407 + 0.913i)13-s − 0.485·14-s − 0.880i·15-s − 0.274·16-s + 1.16·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$403$$    =    $$13 \cdot 31$$ $$\varepsilon$$ = $-0.913 - 0.407i$ motivic weight = $$1$$ character : $\chi_{403} (311, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 403,\ (\ :1/2),\ -0.913 - 0.407i)$ $L(1)$ $\approx$ $0.227580 + 1.06753i$ $L(\frac12)$ $\approx$ $0.227580 + 1.06753i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{13,\;31\}$, $$F_p(T)$$ is a polynomial of degree 2. If $p \in \{13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad13 $$1 + (1.47 - 3.29i)T$$
31 $$1 + iT$$
good2 $$1 - 1.01iT - 2T^{2}$$
3 $$1 + 1.05T + 3T^{2}$$
5 $$1 - 3.24iT - 5T^{2}$$
7 $$1 - 1.79iT - 7T^{2}$$
11 $$1 + 6.09iT - 11T^{2}$$
17 $$1 - 4.79T + 17T^{2}$$
19 $$1 - 1.84iT - 19T^{2}$$
23 $$1 + 7.94T + 23T^{2}$$
29 $$1 + 2.23T + 29T^{2}$$
37 $$1 - 8.64iT - 37T^{2}$$
41 $$1 - 0.0498iT - 41T^{2}$$
43 $$1 - 4.76T + 43T^{2}$$
47 $$1 + 4.00iT - 47T^{2}$$
53 $$1 - 0.588T + 53T^{2}$$
59 $$1 - 3.49iT - 59T^{2}$$
61 $$1 - 12.1T + 61T^{2}$$
67 $$1 + 1.72iT - 67T^{2}$$
71 $$1 - 10.3iT - 71T^{2}$$
73 $$1 - 2.28iT - 73T^{2}$$
79 $$1 - 16.1T + 79T^{2}$$
83 $$1 + 10.2iT - 83T^{2}$$
89 $$1 + 7.99iT - 89T^{2}$$
97 $$1 - 10.1iT - 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

−11.66790571097378381468142422240, −10.92986346737661563086248129139, −10.07239716855035433510475632800, −8.601394198564366576750697875786, −7.82703972707314259283969840703, −6.69753712817983710491653570656, −5.98443139406144540474596435931, −5.57231995976565159605084947453, −3.42914439368563623582348716859, −2.43201092439944989218609452305, 0.74302558891654016651960838591, 2.16197208797998745045430217266, 3.87359674124128814525397028243, 4.94113793092030304031674114951, 5.85379828833843473533549463363, 7.24371390787889943306604764998, 7.983215653127614643728002352023, 9.436846594518341254975733016875, 10.06435272926643452910069214099, 10.88270909596552157995205826476