Properties

 Degree $2$ Conductor $170$ Sign $-0.447 + 0.894i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

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Dirichlet series

 L(s)  = 1 − i·2-s − i·3-s − 4-s + (1 − 2i)5-s − 6-s + i·8-s + 2·9-s + (−2 − i)10-s − 6·11-s + i·12-s − 3i·13-s + (−2 − i)15-s + 16-s + i·17-s − 2i·18-s + 7·19-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.447 − 0.894i)5-s − 0.408·6-s + 0.353i·8-s + 0.666·9-s + (−0.632 − 0.316i)10-s − 1.80·11-s + 0.288i·12-s − 0.832i·13-s + (−0.516 − 0.258i)15-s + 0.250·16-s + 0.242i·17-s − 0.471i·18-s + 1.60·19-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$170$$    =    $$2 \cdot 5 \cdot 17$$ Sign: $-0.447 + 0.894i$ Motivic weight: $$1$$ Character: $\chi_{170} (69, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 170,\ (\ :1/2),\ -0.447 + 0.894i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.596833 - 0.965696i$$ $$L(\frac12)$$ $$\approx$$ $$0.596833 - 0.965696i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
5 $$1 + (-1 + 2i)T$$
17 $$1 - iT$$
good3 $$1 + iT - 3T^{2}$$
7 $$1 - 7T^{2}$$
11 $$1 + 6T + 11T^{2}$$
13 $$1 + 3iT - 13T^{2}$$
19 $$1 - 7T + 19T^{2}$$
23 $$1 - 8iT - 23T^{2}$$
29 $$1 - 5T + 29T^{2}$$
31 $$1 - 5T + 31T^{2}$$
37 $$1 - 8iT - 37T^{2}$$
41 $$1 + 41T^{2}$$
43 $$1 - 4iT - 43T^{2}$$
47 $$1 - 3iT - 47T^{2}$$
53 $$1 + 9iT - 53T^{2}$$
59 $$1 + 5T + 59T^{2}$$
61 $$1 + 3T + 61T^{2}$$
67 $$1 + 2iT - 67T^{2}$$
71 $$1 + 15T + 71T^{2}$$
73 $$1 - 11iT - 73T^{2}$$
79 $$1 + 8T + 79T^{2}$$
83 $$1 + 4iT - 83T^{2}$$
89 $$1 - T + 89T^{2}$$
97 $$1 + 9iT - 97T^{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

−12.56902754620698119956140156272, −11.68846318239185755529641597631, −10.24808373329158249543853928376, −9.756618677118959182678570781437, −8.264859617764210124365604697928, −7.54852750955059439015496471914, −5.69149124039611196796630011913, −4.81233862370992299921138409531, −2.91003541838342100486717755651, −1.24620747171930182430763415435, 2.78849164707539717056843579898, 4.48216253547213087813980636822, 5.58041120481242809972795257783, 6.88711588675284328701798294168, 7.70620778082021457200679001867, 9.132870746615799306542864492406, 10.18243183921051514266936658515, 10.65836645624033867377691971627, 12.19723532688957565209501873946, 13.45019682525548528185759574713