# Properties

 Label 2-17-17.8-c9-0-12 Degree $2$ Conductor $17$ Sign $-0.964 + 0.263i$ Analytic cond. $8.75560$ Root an. cond. $2.95898$ Motivic weight $9$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (16.9 − 16.9i)2-s + (74.3 − 179. i)3-s − 64.3i·4-s + (−2.35e3 − 973. i)5-s + (−1.78e3 − 4.30e3i)6-s + (1.81e3 − 751. i)7-s + (7.59e3 + 7.59e3i)8-s + (−1.27e4 − 1.27e4i)9-s + (−5.64e4 + 2.33e4i)10-s + (6.97e3 + 1.68e4i)11-s + (−1.15e4 − 4.78e3i)12-s − 1.65e5i·13-s + (1.80e4 − 4.35e4i)14-s + (−3.49e5 + 3.49e5i)15-s + 2.90e5·16-s + (−6.54e4 − 3.38e5i)17-s + ⋯
 L(s)  = 1 + (0.750 − 0.750i)2-s + (0.529 − 1.27i)3-s − 0.125i·4-s + (−1.68 − 0.696i)5-s + (−0.562 − 1.35i)6-s + (0.285 − 0.118i)7-s + (0.655 + 0.655i)8-s + (−0.648 − 0.648i)9-s + (−1.78 + 0.739i)10-s + (0.143 + 0.346i)11-s + (−0.160 − 0.0665i)12-s − 1.60i·13-s + (0.125 − 0.302i)14-s + (−1.78 + 1.78i)15-s + 1.10·16-s + (−0.189 − 0.981i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$17$$ Sign: $-0.964 + 0.263i$ Analytic conductor: $$8.75560$$ Root analytic conductor: $$2.95898$$ Motivic weight: $$9$$ Rational: no Arithmetic: yes Character: $\chi_{17} (8, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 17,\ (\ :9/2),\ -0.964 + 0.263i)$$

## Particular Values

 $$L(5)$$ $$\approx$$ $$0.287953 - 2.14363i$$ $$L(\frac12)$$ $$\approx$$ $$0.287953 - 2.14363i$$ $$L(\frac{11}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad17 $$1 + (6.54e4 + 3.38e5i)T$$
good2 $$1 + (-16.9 + 16.9i)T - 512iT^{2}$$
3 $$1 + (-74.3 + 179. i)T + (-1.39e4 - 1.39e4i)T^{2}$$
5 $$1 + (2.35e3 + 973. i)T + (1.38e6 + 1.38e6i)T^{2}$$
7 $$1 + (-1.81e3 + 751. i)T + (2.85e7 - 2.85e7i)T^{2}$$
11 $$1 + (-6.97e3 - 1.68e4i)T + (-1.66e9 + 1.66e9i)T^{2}$$
13 $$1 + 1.65e5iT - 1.06e10T^{2}$$
19 $$1 + (9.72e4 - 9.72e4i)T - 3.22e11iT^{2}$$
23 $$1 + (2.66e5 + 6.42e5i)T + (-1.27e12 + 1.27e12i)T^{2}$$
29 $$1 + (-3.90e6 - 1.61e6i)T + (1.02e13 + 1.02e13i)T^{2}$$
31 $$1 + (-1.44e6 + 3.50e6i)T + (-1.86e13 - 1.86e13i)T^{2}$$
37 $$1 + (-4.54e6 + 1.09e7i)T + (-9.18e13 - 9.18e13i)T^{2}$$
41 $$1 + (9.08e6 - 3.76e6i)T + (2.31e14 - 2.31e14i)T^{2}$$
43 $$1 + (-2.50e7 - 2.50e7i)T + 5.02e14iT^{2}$$
47 $$1 - 5.08e7iT - 1.11e15T^{2}$$
53 $$1 + (1.09e7 - 1.09e7i)T - 3.29e15iT^{2}$$
59 $$1 + (6.57e7 + 6.57e7i)T + 8.66e15iT^{2}$$
61 $$1 + (-1.88e8 + 7.81e7i)T + (8.26e15 - 8.26e15i)T^{2}$$
67 $$1 + 5.35e7T + 2.72e16T^{2}$$
71 $$1 + (5.11e7 - 1.23e8i)T + (-3.24e16 - 3.24e16i)T^{2}$$
73 $$1 + (-4.04e7 - 1.67e7i)T + (4.16e16 + 4.16e16i)T^{2}$$
79 $$1 + (5.24e7 + 1.26e8i)T + (-8.47e16 + 8.47e16i)T^{2}$$
83 $$1 + (2.90e8 - 2.90e8i)T - 1.86e17iT^{2}$$
89 $$1 + 2.89e8iT - 3.50e17T^{2}$$
97 $$1 + (-2.61e8 - 1.08e8i)T + (5.37e17 + 5.37e17i)T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−15.93057303684148063986171423221, −14.36914964146034846710376232214, −12.85460873432139384452542579637, −12.42375905284696191099501901275, −11.22753936854724217803514745590, −8.204007792835576401374598022231, −7.58359373353851393849781788077, −4.56040358040895556196564079798, −2.90523523835195394168116930723, −0.865662797836297744395558625475, 3.71190566556529794932326163624, 4.49787069465201684461842647350, 6.81494326405852293520630822376, 8.500775357410407481909227166811, 10.40163861459722009397813064815, 11.72480289657472913222075801096, 14.05783037491342855900326042171, 14.95475060654385426128675603168, 15.60407251209329335536819402896, 16.47428015261393061355685060853