# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 5 \cdot 67$ Sign $0.990 + 0.141i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 − i·3-s + (2.21 + 0.315i)5-s − 0.0918i·7-s − 9-s − 3.06·11-s + 5.35i·13-s + (0.315 − 2.21i)15-s − 2.02i·17-s + 7.88·19-s − 0.0918·21-s − 5.50i·23-s + (4.80 + 1.39i)25-s + i·27-s − 2.34·29-s − 1.86·31-s + ⋯
 L(s)  = 1 − 0.577i·3-s + (0.990 + 0.141i)5-s − 0.0347i·7-s − 0.333·9-s − 0.923·11-s + 1.48i·13-s + (0.0814 − 0.571i)15-s − 0.492i·17-s + 1.80·19-s − 0.0200·21-s − 1.14i·23-s + (0.960 + 0.279i)25-s + 0.192i·27-s − 0.435·29-s − 0.335·31-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4020$$    =    $$2^{2} \cdot 3 \cdot 5 \cdot 67$$ $$\varepsilon$$ = $0.990 + 0.141i$ motivic weight = $$1$$ character : $\chi_{4020} (1609, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4020,\ (\ :1/2),\ 0.990 + 0.141i)$ $L(1)$ $\approx$ $2.245970564$ $L(\frac12)$ $\approx$ $2.245970564$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;5,\;67\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;67\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
3 $$1 + iT$$
5 $$1 + (-2.21 - 0.315i)T$$
67 $$1 + iT$$
good7 $$1 + 0.0918iT - 7T^{2}$$
11 $$1 + 3.06T + 11T^{2}$$
13 $$1 - 5.35iT - 13T^{2}$$
17 $$1 + 2.02iT - 17T^{2}$$
19 $$1 - 7.88T + 19T^{2}$$
23 $$1 + 5.50iT - 23T^{2}$$
29 $$1 + 2.34T + 29T^{2}$$
31 $$1 + 1.86T + 31T^{2}$$
37 $$1 - 6.88iT - 37T^{2}$$
41 $$1 + 5.27T + 41T^{2}$$
43 $$1 + 6.19iT - 43T^{2}$$
47 $$1 - 3.51iT - 47T^{2}$$
53 $$1 + 1.50iT - 53T^{2}$$
59 $$1 - 7.85T + 59T^{2}$$
61 $$1 - 12.9T + 61T^{2}$$
71 $$1 - 12.3T + 71T^{2}$$
73 $$1 - 6.48iT - 73T^{2}$$
79 $$1 - 8.72T + 79T^{2}$$
83 $$1 - 3.43iT - 83T^{2}$$
89 $$1 - 4.92T + 89T^{2}$$
97 $$1 + 3.65iT - 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

−8.482642456440773516194242393028, −7.53912559928598752586365227139, −6.91063765796460228030642386795, −6.38569337090952876398296511585, −5.33846105950175498337387092027, −5.01676815549585183268366425966, −3.70685672979882029831063929323, −2.64433019345373162613132955043, −2.04770441183105429666464355002, −0.930621101673477384771255784771, 0.812959965645074319168758609190, 2.09670001557287879230279538323, 3.05054759036481549937507292321, 3.68378498917271482507744842828, 5.08226337726605185505685732733, 5.42326641902881677285774995897, 5.86588320455952925695819276656, 7.09954530035153848008023098724, 7.79537312366796050103623129187, 8.467201346295137872455597357499