# Properties

 Degree 2 Conductor $13 \cdot 31$ Sign $0.832 + 0.554i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2·3-s + 2·4-s − 4i·5-s + 2i·7-s + 9-s − i·11-s + 4·12-s + (−2 + 3i)13-s − 8i·15-s + 4·16-s − 2·17-s + 2i·19-s − 8i·20-s + 4i·21-s − 4·23-s + ⋯
 L(s)  = 1 + 1.15·3-s + 4-s − 1.78i·5-s + 0.755i·7-s + 0.333·9-s − 0.301i·11-s + 1.15·12-s + (−0.554 + 0.832i)13-s − 2.06i·15-s + 16-s − 0.485·17-s + 0.458i·19-s − 1.78i·20-s + 0.872i·21-s − 0.834·23-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$403$$    =    $$13 \cdot 31$$ $$\varepsilon$$ = $0.832 + 0.554i$ motivic weight = $$1$$ character : $\chi_{403} (311, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 403,\ (\ :1/2),\ 0.832 + 0.554i)$ $L(1)$ $\approx$ $2.17962 - 0.659937i$ $L(\frac12)$ $\approx$ $2.17962 - 0.659937i$ $L(\frac{3}{2})$ 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 + (2 - 3i)T$$
31 $$1 - iT$$
good2 $$1 - 2T^{2}$$
3 $$1 - 2T + 3T^{2}$$
5 $$1 + 4iT - 5T^{2}$$
7 $$1 - 2iT - 7T^{2}$$
11 $$1 + iT - 11T^{2}$$
17 $$1 + 2T + 17T^{2}$$
19 $$1 - 2iT - 19T^{2}$$
23 $$1 + 4T + 23T^{2}$$
29 $$1 - 8T + 29T^{2}$$
37 $$1 - 3iT - 37T^{2}$$
41 $$1 - 4iT - 41T^{2}$$
43 $$1 - 4T + 43T^{2}$$
47 $$1 - 2iT - 47T^{2}$$
53 $$1 + 4T + 53T^{2}$$
59 $$1 + 6iT - 59T^{2}$$
61 $$1 - 10T + 61T^{2}$$
67 $$1 + 16iT - 67T^{2}$$
71 $$1 - 6iT - 71T^{2}$$
73 $$1 - 7iT - 73T^{2}$$
79 $$1 + 16T + 79T^{2}$$
83 $$1 + 12iT - 83T^{2}$$
89 $$1 - 15iT - 89T^{2}$$
97 $$1 + 2iT - 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

−11.48824188220044485522628631094, −9.950788799957586492306938655011, −9.148093643779623950381364834693, −8.424697193241376930589622367273, −7.87332931386543365109413609431, −6.42619710812602113355812489125, −5.34892375760208459795988448951, −4.15389208039854490017879517281, −2.67639293080402667122026786589, −1.67317628191153079059259795335, 2.33898214138781229335548978487, 2.87913545534096285272776893375, 3.91813201613630258886152375650, 5.93747167203652307881585637367, 7.03246694791877821157741815824, 7.39411504380701027906515046948, 8.330891792706934249857488501637, 9.865101921932921833798347143972, 10.40650301407362823743526944467, 11.11640169391882038432642624454