# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 59$ Sign $-0.964 - 0.264i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + (0.869 + 1.49i)3-s + 4-s + 1.66i·5-s + (−0.869 − 1.49i)6-s − 3.57·7-s − 8-s + (−1.48 + 2.60i)9-s − 1.66i·10-s − 4.82·11-s + (0.869 + 1.49i)12-s − 0.418i·13-s + 3.57·14-s + (−2.48 + 1.44i)15-s + 16-s − 0.742i·17-s + ⋯
 L(s)  = 1 − 0.707·2-s + (0.502 + 0.864i)3-s + 0.5·4-s + 0.742i·5-s + (−0.355 − 0.611i)6-s − 1.35·7-s − 0.353·8-s + (−0.495 + 0.868i)9-s − 0.524i·10-s − 1.45·11-s + (0.251 + 0.432i)12-s − 0.116i·13-s + 0.956·14-s + (−0.642 + 0.372i)15-s + 0.250·16-s − 0.180i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$354$$    =    $$2 \cdot 3 \cdot 59$$ $$\varepsilon$$ = $-0.964 - 0.264i$ motivic weight = $$1$$ character : $\chi_{354} (353, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 354,\ (\ :1/2),\ -0.964 - 0.264i)$ $L(1)$ $\approx$ $0.0779952 + 0.579371i$ $L(\frac12)$ $\approx$ $0.0779952 + 0.579371i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;59\}$, $$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + (-0.869 - 1.49i)T$$
59 $$1 + (-5.47 + 5.38i)T$$
good5 $$1 - 1.66iT - 5T^{2}$$
7 $$1 + 3.57T + 7T^{2}$$
11 $$1 + 4.82T + 11T^{2}$$
13 $$1 + 0.418iT - 13T^{2}$$
17 $$1 + 0.742iT - 17T^{2}$$
19 $$1 - 1.24T + 19T^{2}$$
23 $$1 - 1.25T + 23T^{2}$$
29 $$1 + 4.78iT - 29T^{2}$$
31 $$1 - 10.4iT - 31T^{2}$$
37 $$1 - 6.02iT - 37T^{2}$$
41 $$1 - 9.24iT - 41T^{2}$$
43 $$1 + 3.58iT - 43T^{2}$$
47 $$1 + 8.39T + 47T^{2}$$
53 $$1 - 9.66iT - 53T^{2}$$
61 $$1 - 9.08iT - 61T^{2}$$
67 $$1 - 13.0iT - 67T^{2}$$
71 $$1 + 4.01iT - 71T^{2}$$
73 $$1 + 10.6iT - 73T^{2}$$
79 $$1 - 3.55T + 79T^{2}$$
83 $$1 + 2.87T + 83T^{2}$$
89 $$1 - 16.0T + 89T^{2}$$
97 $$1 + 11.5iT - 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

−11.54385675600396742251036226615, −10.37770811813735534936973403293, −10.25663725548389229507880189388, −9.280679739517931336114181371808, −8.296828760314077096337823107205, −7.31573079607611625570094090941, −6.27860994755856958010651042599, −4.99580644492729200764942043894, −3.23803952970376798417246378690, −2.75059940896852471278931911046, 0.44038283552706877865809875496, 2.28211974081840585064491615231, 3.43316960280639529696700690842, 5.38803806778467572010911440168, 6.47320942826670643312044386056, 7.42237729760466416561537149257, 8.252311428157221318248102208910, 9.151214928524464038968547652122, 9.831905273571093651721480532359, 10.95581581726219086221310924872