# Properties

 Degree 2 Conductor $2^{2} \cdot 17$ Sign $0.615 + 0.788i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 − i·2-s − 4-s + (−1 + i)5-s + i·8-s − i·9-s + (1 + i)10-s + 16-s − 17-s − 18-s + (1 − i)20-s − i·25-s + (1 − i)29-s − i·32-s + i·34-s + i·36-s + (−1 + i)37-s + ⋯
 L(s)  = 1 − i·2-s − 4-s + (−1 + i)5-s + i·8-s − i·9-s + (1 + i)10-s + 16-s − 17-s − 18-s + (1 − i)20-s − i·25-s + (1 − i)29-s − i·32-s + i·34-s + i·36-s + (−1 + i)37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$68$$    =    $$2^{2} \cdot 17$$ $$\varepsilon$$ = $0.615 + 0.788i$ motivic weight = $$0$$ character : $\chi_{68} (55, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 68,\ (\ :0),\ 0.615 + 0.788i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.4210945963$$ $$L(\frac12)$$ $$\approx$$ $$0.4210945963$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;17\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + iT$$
17 $$1 + T$$
good3 $$1 + iT^{2}$$
5 $$1 + (1 - i)T - iT^{2}$$
7 $$1 - iT^{2}$$
11 $$1 - iT^{2}$$
13 $$1 + T^{2}$$
19 $$1 + T^{2}$$
23 $$1 - iT^{2}$$
29 $$1 + (-1 + i)T - iT^{2}$$
31 $$1 + iT^{2}$$
37 $$1 + (1 - i)T - iT^{2}$$
41 $$1 + (-1 - i)T + iT^{2}$$
43 $$1 + T^{2}$$
47 $$1 - T^{2}$$
53 $$1 - T^{2}$$
59 $$1 + T^{2}$$
61 $$1 + (1 + i)T + iT^{2}$$
67 $$1 - T^{2}$$
71 $$1 + iT^{2}$$
73 $$1 + (-1 + i)T - iT^{2}$$
79 $$1 - iT^{2}$$
83 $$1 + T^{2}$$
89 $$1 + T^{2}$$
97 $$1 + (1 - i)T - iT^{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

−14.84625762244638749011823645362, −13.73708928432188607873060131214, −12.36784286101715037183258870678, −11.54553781733229543264173850657, −10.67475820275141773886786578035, −9.401820822599483704682833492791, −8.063910698916515610767798864707, −6.52495137986341477590907454457, −4.29878395807902601847606510864, −3.03544157255568814852703229915, 4.22564061217201658866752193569, 5.27019738107696468255483049268, 7.09193503008472694559106097842, 8.208046929617629467128032615792, 8.986147440681671677679816455417, 10.68372419403733204059333291826, 12.21051665090741868031115105838, 13.15884627493339383429814612152, 14.21237496700354657276706814216, 15.59461919004984204952321323569